essay on application of differential equations

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

Essay about the art and applications of differential equations?

I teach a high school calculus class. We've worked through the standard derivatives material, and I incorporated a discussion of antiderivatives throughout. I've introduced solving "area under a curve" problems as solving differential equations. Since it's easy to see the rate at which area is accumulating (the height of the function), we can write down a differential equation, take an antiderivative to find the area function, and solve for the constant.

Anyhow, I find myself wanting to share with my students what differential equations are all about. I don't know so much about them, but I have a sense that they are both a beautiful pure mathematics subject and a subject that has many, many applications.

I'm hoping to hear suggestions for an essay or a chapter from a book--something like 3 to 10 pages--that would discuss differential equations as a subject, give some interesting examples, and point to some applications. Any ideas?

reference-request
ordinary-differential-equations

$\begingroup$ The first pages of Hirsch & Smale's "Differential Equations, Dynamical Systems & An Introduction to Chaos" might be useful. It is fairly easy and gives some motivated, applied and qualitative examples $\endgroup$ – Bruno Stonek Commented Feb 9, 2012 at 0:00
1 $\begingroup$ If you don't mind internet resources, take a look at Alan Rendall's blog . He writes about mostly ordinary and partial differential equations in their applications to mathematical biology and general relativity. $\endgroup$ – Willie Wong Commented Feb 9, 2012 at 0:58
$\begingroup$ Look at math.stackexchange.com/questions/107264/… it may be a real eye-opener. $\endgroup$ – Artes Commented Feb 10, 2012 at 1:52
1 $\begingroup$ Try Arnold's essay Evolution Processes and Ordinary Differential Equations , written exactly for high school students. $\endgroup$ – Marius Kempe Commented Sep 2, 2014 at 17:45
1 $\begingroup$ Here's a little big hobby project I started a few weeks ago along those lines (pun intended): slopefield.wordpress.com – I'm not sure if I'm ever going to finish building something "final" out of it, but I think you're going to find a lot of material for an essay in there, if you're still interested. $\endgroup$ – Bogdan Stăncescu Commented Mar 9, 2016 at 21:38

2 Answers 2

Differential equations is a rather immense subject. In spite of the risk of overwhelming you with the amount of information, I recommend looking in the Princeton Companion to Mathematics , from which the relevant sections are (page numbers are within parts)

Section I.3.5.4 for an introductory overview
Section I.4.1.5
Section III.23 on differential equations describing fluids (including the Navier-Stokes equation which is the subject of one of the Millennium problems)
Section III.36 especially on the heat equation and its relation to various topics in mathematical physics and finance
Section III.51 on wave phenomenon
Section IV.12 on partial differential equations as a branch of mathematics
Section IV.14 on dynamical systems and ordinary differential equations
Section V.36 on the three body problem
Section VII.2 on mathematical biology

Some of these material may be too advanced or too detailed for your purposes. But they may on the other hand provide keywords and phrases for you to improve your search.

I strongly recommend to read a review paper with many interesting references therein :

PDE as a Unified Subject by Sergiu Klainerman.

An essay on partial differential equations written by a leading expert in the field, for anyone attemping to know more on the subject as well as to those who would like to get a grasp of interactions between Mathematics and Physics.

Another very interesting article dealing with certain aspects of differential equations (to some extent) and teaching is V.I. Arnold's On teaching mathematics (it may be worth to read if you are a teacher). A bit more detailed but still very clear is his essay “Mathematics and physics: mother and daughter or sisters?" (check another sites if you can't download it).

$\begingroup$ I should add that Arnold's introduction to ODE is great reading (he tried to keep the examples very simple but powerful with emphasis on flow, topology, singularity and so on). $\endgroup$ – Raymond Manzoni Commented Feb 12, 2012 at 21:44
$\begingroup$ Yes, I can say the same of his another books like "Mathematical Methods of Classical Mechanics" and "Geometrical Methods in the Theory of differential equations". $\endgroup$ – Artes Commented Feb 12, 2012 at 22:01

You must log in to answer this question.

Not the answer you're looking for browse other questions tagged calculus reference-request ordinary-differential-equations ..

Upcoming Events
2024 Community Moderator Election ends in 6 days
Featured on Meta
We've made changes to our Terms of Service & Privacy Policy - July 2024
Bringing clarity to status tag usage on meta sites
Upcoming Moderator Election
2024 Community Moderator Election

Hot Network Questions

Is Apex Trigger considers two updates on different records from two users into a single transaction?
A command for typing ʍ in LaTeX
John 11 and The Sanhedrin Council
What is a word/phrase that best describes a "blatant disregard or neglect" for something, but with the connotation of that they should have known?
Would several years of appointment as a lecturer hurt you when you decide to go for a tenure-track position later on?
What is the soteriological significance of Hebrews 10:1 in the context of God's Redemption Plan?
Why does Air Force Two lack a tail number?
Has the application of a law ever being appealed anywhere due to the lawmakers not knowing what they were voting/ruling?
Confused about topographic data in base map using QGIS
Cutting a 27×27 square into incomparable rectangles
Can someone help me identify this plant?
How are USB-C cables so thin?
What role does the lower bound play in the statement of Savitch's Theorem?
Why is Excel not counting time with COUNTIF?
Sums of X*Y chunks of the nonnegative integers
With 42 supernovae in 37 galaxies, how do we know SH0ES results is robust?
Word to classify what powers a god is associated with?
The minimal Anti-Sudoku
Do space stations have anything that big spacecraft (such as the Space Shuttle and SpaceX Starship) don't have?
What happened to the genealogical records of the tribe of Judah and the house of David after Jerusalem was destroyed by the Romans in 70 A.D
Are there rules of when there is linking-sound compound words?
can a CPU once removed retain information that poses a security concern?
Erase the loops
What is the meaning of "Exit, pursued by a bear"?

$essay on application of differential equations$

The Open University
Accessibility hub
Guest user / Sign out
Study with The Open University

My OpenLearn Profile

Personalise your OpenLearn profile, save your favourite content and get recognition for your learning

About this free course

Become an ou student, share this free course.

Start this free course now. Just create an account and sign in. Enrol on the course to track your learning.

Introduction to differential equations

Introduction.

Please note: a Statement of Participation is not issued for this course.

This free OpenLearn course, Introduction to differential equations, is an extract from the Open University module MST125 Essential mathematics 2 [ Tip: hold Ctrl and click a link to open it in a new tab. ( Hide tip ) ] . The module builds on mathematical ideas introduced in MST124 Essential mathematics 1 , covering a wide range of topics from different areas of mathematics. The practical application to problems provides a firm foundation for further studies in mathematics and other mathematically rich subjects such as physics and engineering. Topics covered include mathematical typesetting, number theory, conics, statics, geometric transformations, calculus, differential equations, mathematical language and proof, dynamics, eigenvalues and eigenvectors and combinatorics. It also helps develop the abilities to study mathematics independently, to solve mathematical problems and to communicate mathematics.

Introduction to differential equations consists of material from MST125 Unit 8, Differential equations and has six sections in total. Sections 5 and 6 are considerably shorter than the other four sections. You should set aside about four hours to study each of the first four sections and about two hours for the remaining two sections; the whole extract should take about 18 hours to study. The extract is a small part (around 8%) of a large module that is studied over eight months, and so can give only an approximate indication of the level and content of the full module.

The extract, which contains an introduction to differential equations, is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in MST125. A few techniques and definitions taught in earlier units in MST125 are present in the extract without explanation, therefore a fluency with algebra and basic calculus is essential for this extract.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the MST125 software or to video material (although please note that the PDF may contain references to other parts of MST125). In this extract, some illustrations have also been removed due to copyright restrictions.

Regrettably, mathematical and statistical content in PDF form is not accessible using a screenreader, and you may need additional help to read these documents.

Differential equations are any equations that include derivatives. They arise in many situations in mathematics, physics, chemistry, engineering, biology, economics and finance. Three types of first-order differential equations are considered.

Section 1 introduces you to equations that can be solved by direct integration.

Section 2 introduces the method of separation of variables for solving differential equations.

Section 3 looks at applications of differential equations for solving real world problems including variations in the size of a population over time and radioactive decay.

Section 4 introduces you to the integrating factor method for solving linear differential equations.

The final two sections summarise and revise the methods introduced in the previous sections and describe various other approaches to finding solutions of first-order differential equations and to understanding the behaviour of the solutions.

essay on application of differential equations

500 Examples and Problems of Applied Differential Equations

© 2019
Ravi P. Agarwal 0 ,
Simona Hodis 1 ,
Donal O’Regan 2

Department of Mathematics, Texas A&M University–Kingsville, Kingsville, USA

You can also search for this author in PubMed Google Scholar

Department of Mathematics, National University of Ireland, Galway, Ireland

Highlights an unprecedented number of real-life applications of differential equations and systems
Includes problems in biomathematics, finance, engineering, physics, and even societal ones like rumors and love
Includes selected challenges to motivate further research in this field

Part of the book series: Problem Books in Mathematics (PBM)

38k Accesses

9 Citations

This is a preview of subscription content, log in via an institution to check access.

Access this book

Subscribe and save.

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime
Available as EPUB and PDF
Read on any device
Instant download
Own it forever
Compact, lightweight edition
Dispatched in 3 to 5 business days
Free shipping worldwide - see info
Durable hardcover edition

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

About this book

This book highlights an unprecedented number of real-life applications of differential equations together with the underlying theory and techniques. The problems and examples presented here touch on key topics in the discipline, including first order (linear and nonlinear) differential equations, second (and higher) order differential equations, first order differential systems, the Runge–Kutta method, and nonlinear boundary value problems. Applications include growth of bacterial colonies, commodity prices, suspension bridges, spreading rumors, modeling the shape of a tsunami, planetary motion, quantum mechanics, circulation of blood in blood vessels, price-demand-supply relations, predator-prey relations, and many more.

Upper undergraduate and graduate students in Mathematics, Physics and Engineering will find this volume particularly useful, both for independent study and as supplementary reading. While many problems can be solved at the undergraduate level, a numberof challenging real-life applications have also been included as a way to motivate further research in this vast and fascinating field.

General Systems of Differential Equations

Ordinary differential equations

Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

first order linear differential equations
second and higher order differential equations
first order nonlinear differential equations
power series solutions
first order differential systems
numerical methods
stability theory
linear boundary value problems
nonlinear boundary value problems
Runge–Kutta method
Fourier method
ordinary differential equations
partial differential equations

Table of contents (10 chapters)

Front matter, first-order linear differential equations.

Ravi P. Agarwal, Simona Hodis, Donal O’Regan

Some First-Order Nonlinear Differential Equations

Second- and higher order differential equations, power series solutions, systems of first-order differential equations, runge–kutta method, stability theory, linear boundary value problems, nonlinear boundary value problems, correction to: systems of first-order differential equations, back matter.

“The book provides an excellent collection of ideas to spice up a lecture on differential equations with an analytical approach and thus to increase the motivation of students.” (Volker H. Schulz, SIAM Review, Vol. 62 (3), 2020)

Authors and Affiliations

Department of mathematics, texas a&m university–kingsville, kingsville, usa.

Ravi P. Agarwal, Simona Hodis

Donal O’Regan

About the authors

Bibliographic information.

Book Title : 500 Examples and Problems of Applied Differential Equations

Authors : Ravi P. Agarwal, Simona Hodis, Donal O’Regan

Series Title : Problem Books in Mathematics

DOI : https://doi.org/10.1007/978-3-030-26384-3

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Hardcover ISBN : 978-3-030-26383-6 Published: 19 December 2019

Softcover ISBN : 978-3-030-26386-7 Published: 21 January 2021

eBook ISBN : 978-3-030-26384-3 Published: 24 September 2019

Series ISSN : 0941-3502

Series E-ISSN : 2197-8506

Edition Number : 1

Number of Pages : IX, 388

Number of Illustrations : 81 b/w illustrations, 3 illustrations in colour

Topics : Ordinary Differential Equations , Difference and Functional Equations , Partial Differential Equations , Sequences, Series, Summability , Numerical Analysis

Publish with us

Policies and ethics

Find a journal
Track your research

Before you go, check this out!

We have lots more on the site to show you. You've only seen one page. Check out this post which is one of the most popular of all time.

The Impact of Differential Equations in Our Everyday Lives (9 examples!)

Have you ever wondered how scientists and engineers manage to predict the behavior of complex systems, including forecasting the weather, tracking the stock market, or even the universe itself? Much of their ability to do so can be traced back to the fascinating world of differential equations.

Differential equations are powerful equations that provide a mathematical model to describe the fundamental laws governing a given system. Additionally, differential equations enable us to understand, control, and predict the world around us with remarkable accuracy.

In this blog post, I will explore some real-life applications of differential equations that demonstrate their importance and connection to various aspects of our daily lives.

If you wonder whether differential equations are hard , I wrote a whole article where I share helpful tips to ace differential equations.

Table of Contents

What Are Differential Equations?

In Mathematics, differential equations are equations with one or more function derivatives. The derivative of the function is expressed by dy/dx. In other words, a differential equation is any equation containing one or more function derivatives. Generally, differential equations describe relationships that involve rates of change.

The vast and pervasive influence of differential equations in our daily lives goes beyond what we have covered in this journey, encompassing diverse fields such as art, social sciences, and more.

The beauty of these equations lies in their ability to turn seemingly intangible concepts into quantifiable relationships and models, paving the way for advancements in various branches of science, engineering, and even art.

To appreciate differential equations is to appreciate their power in deciphering the world’s complexities, unlocking the secrets of our universe, and, ultimately, transforming the world for the better.

The table below shows multiple representations of differential equations.

The Impact of Differential Equations in Our Everyday Lives

Differential Equations Applications in Real Life

The applications of differential equations go far beyond the realms of academia, touching every aspect of our lives, from biology to the engineering around us and even to the behavior of the economy.

Differential equations are used in various disciplines, from biology, economics, physics, chemistry, and engineering. They can describe exponential growth and decay, species’ population growth, or the change in investment return over time.

Here are the most common differential equations applications in real life.

1- Weather Forecasting

One of the most immediate applications of differential equations that comes to mind is in the field of weather forecasting. Meteorologists rely on mathematical models derived from differential equations to predict the movements and interactions of temperature, pressure, and moisture in our atmosphere.

Differential equations models allow them to make estimates of future weather conditions that inform everything from our daily commutes to major decisions made in the agricultural and aviation industries.

As technology and computing power continue to advance, our ability to generate more accurate short and long-term weather forecasts will keep improving, all thanks to the power of differential equations.

2- Modeling Ecology

Biologists use differential equations to understand how populations of plants, animals, and bacteria grow and interact with one another. The well-known logistic equation, a first-order differential equation, helps to model the growth of a population by accounting for factors such as birth rates, death rates, and ecological constraints like resource availability.

The Lorenz equations , a system of three coupled differential equations, model the role of chaos in a variety of natural phenomena, such as predator-prey interactions and the behavior of atmospheric systems.

3- Modeling Population Dynamics

Another realm where differential equations play a crucial role is in predicting the growth and decline of populations, be they bacterial colonies or entire nations of people.

Differential equations models can be helpful in understanding how factors such as birth rates, death rates, and migration influence the size and distribution of populations over time.

One of the most applications of differential equations is the Malthusian Law of population growth dp/dt = rp illustrates how the population (p) varies with respect to time. The constant r will vary depending on the species.

Moreover, differential equation models can also be applied to the study of disease spreading, such as during pandemic outbreaks, which provides insights into how public health officials can implement measures to control the spread of diseases and mitigate their impacts on society.

Differential equations Applications In Population Dynamics

4- Physics and Engineering

Both classical and modern physics rely heavily on differential equations to describe the laws that govern the universe. From Newton’s laws of motion to Maxwell’s equations of electromagnetism , these mathematical models are essential tools in revealing how physical quantities such as position, velocity, and acceleration change over time.

Additionally, engineers use differential equations to design and analyze a wide variety of systems, ranging from chemical reactors to electrical circuits. By providing a quantitative framework to predict and optimize the performance of these systems, differential equations serve as a foundation upon which much of our modern world is built.

5- Economics and Finance

One might be surprised to learn that differential equations have a significant role to play in finance and economics. The Black-Scholes equation , a partial differential equation that originated from mathematical physics, provides a framework for valuing stock options and pricing financial derivatives.

Such a framework is invaluable in the management of risk and the optimization of investment strategies. Differential equations models are frequently used to determine the optimal investment strategies, forecast changes in commodity prices, and generally evaluate the efficiency and stability of economic systems.

Furthermore, in analyzing the movement of stock prices, for example, researchers often employ stochastic differential equations to account for the inherent uncertainty and random fluctuations that characterize financial markets.

In economics, differential equations are used to model and analyze the behavior of several variables, such as consumption, production, and employment, thereby helping policymakers to make informed decisions and design optimal policies for economic growth and stability.

Differential equations Applications In Economics and Finance

6- Environmental Science

Differential equations are also essential tools in revealing the complex interactions between organisms and their environments. Environmental ists use these models to study predator-prey relationships, competition among species, and the effects of various environmental factors on population dynamics.

By informing conservation practices, these mathematical models guide the sustainable management of ecosystems and improve our ability to preserve biodiversity.

Additionally, environmental scientists deploy differential equations to analyze the transport and fate of pollutants in air, water, and soil, which is an essential aspect of ensuring the long-term health and well-being of our planet.

Differential equations Applications In Environmental Science

7- Medicine and Healthcare

Differential equations also find widespread applications in the world of medicine and healthcare. The modeling of biological systems allows researchers to study disease progression, design therapies, and predict patient outcomes.

For instance, the renowned SIR Model for Spread of Disease , a set of differential equations, represents the dynamics of infectious diseases, dissecting the relationships between susceptible, infected, and recovered individuals within a population.

Furthermore, pharmacokinetics and pharmacodynamics use differential equations to study the distribution and action of drugs within our bodies. This knowledge aids in drug development, clinical trials, and the treatment of patients.

Differential equations Applications In Medicine and Healthcare

8- Modeling Climate Change

Our changing climate is one of the most pressing issues facing our planet, and understanding the complex environmental processes is crucial to addressing this challenge. Differential equations assist in modeling and predicting climate change through the study of atmospheric, oceanic, and terrestrial processes.

For example, the use of partial differential equations allows scientists to develop complex numerical models of the Earth’s climate system, accounting for factors such as radiation, cloud formation, wind patterns, and heat exchange across various compartments.

Differential equations models provide critical insights into the likely impact of anthropogenic activities on our climate and help inform policies on mitigating the effects of climate change.

Differential equations Applications In Modeling Climate Change

9- Applied Engineering Studies

Differential equations are fundamental in developing engineering models and understanding physical phenomena. In engineering, differential equations are used to solve problems related to fluid dynamics, heat transfer, stress analysis, and more.

For example, the Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluids, allowing engineers to design and analyze the performance of pumps, turbines, pipelines, and airfoils.

In physics, Maxwell’s equations, a set of partial differential equations, govern the behavior of electric and magnetic fields, leading to significant advancements in our understanding of electromagnetism and the development of modern technologies such as power generation, telecommunications, and electronics.

In Conclusion

While differential equations may seem like an abstract and obscure concept from the realm of mathematics, their applications in real-life scenarios are numerous, tangible, and truly transformative.

From predicting and shaping the world’s weather, understanding the intricacies of economics, and preserving our planet’s ecosystems, I believe that these mathematical models are critical tools that empower us to interpret and navigate the complexity of life.

With every new discovery and application, the power of differential equations will continue to revolutionize the way we live, work, and thrive.

So the next time you check your local weather forecast, remember that behind every prediction lies a fascinating world of mathematics that plays an indispensable role in our everyday lives.

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

Share this article

Latest updates.

Ways To Improve Learning Outcomes: Learn Tips & Tricks

Visual Learning Style for Students: Pros and Cons

NCERT Books for Class 6 Social Science 2024 – Download PDF

CBSE Syllabus for Class 9 Social Science 2023-24: Download PDF

CBSE Syllabus for Class 8 Maths 2024: Download PDF

NCERT Books for Class 6 Maths 2025: Download Latest PDF

CBSE Class 10 Study Timetable 2024 – Best Preparation Strategy

CBSE Class 10 Syllabus 2025 – Download PDF

CBSE Syllabus for Class 11 2025: Download PDF

NCERT Solutions for Class 7 Science Chapter 16 Water – A Precious Resource

Application of Differential Equations: Definition, Types, Examples

Applications of Differential Equations: A differential equation, also abbreviated as D.E., is an equation for the unknown functions of one or more variables. It relates the values of the function and its derivatives. Differential equations have applications in various fields of Science like Physics (dynamics, thermodynamics, heat, fluid mechanics, and electromagnetism), Chemistry (rate of chemical reactions, physical chemistry, and radioactive decay), Biology (growth rates of bacteria, plants and other organisms) and Economics (economic growth rate, and population growth rate). Having said that, almost all modern scientific investigations involve differential equations.

Differential Equation: Overview

The equation that involves independent variables, dependent variables and their derivatives is called a differential equation.

There are two types of differential equations:

Ordinary Differential Equations
Partial Differential Equations

Applications of Differential Equations in Real Life

The applications of differential equations in real life are as follows:

In Physics:

Study the movement of an object like a pendulum
Study the movement of electricity
To represent thermodynamics concepts

In Medicine:

Graphical representations of the development of diseases

In Mathematics:

Describe mathematical models such as:

population explosion
radioactive decay

Geometrical applications:

To find the:

slope of a tangent
equation of tangent and normal
length of tangent and normal
length of sub-tangent and sub-normal

Physical application:

We can calculate

acceleration

Applications of First-order Differential Equations

The applications of the First-order differential equations are as follows:

What is an ODE?

An ordinary differential equation, or ODE, is a differential equation in which the dependent variable is a function of the independent variable. The highest order derivative in the differential equation is called the order of the differential equation.

First-Order ODE

The first-order differential equation is given by

$\frac{{dy}}{{dx}} = f(x,\,y)$

$x$ is the independent variable

$y$ is the dependent variable.

It has only the first-order derivative $\frac{{dy}}{{dx}}$. Hence, the order is $1$.

First-order differential equations have a wide range of applications.

Growth and Decay

Let $N(t)$ denote the amount of substance (or population) that is growing or decaying. If we assume that the time rate of change of this amount of substance, $\frac{{dN}}{{dt}}$, is proportional to the amount of substance present, then

$\frac{{dN}}{{dt}} = kN$, or $\frac{{dN}}{{dt}} – kN = 0$

Where, $k$ is the constant of proportionality.

Here, we assume that $N(t)$ is a differentiable, continuous function of time.

Newton’s Cooling Law

Newton’s law of cooling and heating, states that the rate of change of the temperature in the body, $\frac{{dT}}{{dt}}$, is proportional to the temperature difference between the body and its medium. Newton’s law of cooling can be formulated as

$\frac{{dT}}{{dt}} = – k\left( {T – {T_m}} \right)$

$ \Rightarrow \frac{{dT}}{{dt}} + kT = k{T_m}$

Where $k$ is a positive constant of proportionality.

Applications of Second-order Differential Equations

The applications of second-order differential equations are as follows:

Second-Order ODE

The second-order differential equation is given by

${y^{\prime \prime }} + p(x){y^\prime } + q(x)y = f(x)$

$p\left( x \right)$ and $q\left( x \right)$ are either constant or function of $x$.

The highest order derivative is $\frac{{{d^2}y}}{{d{x^2}}}$. Hence, the order is $2$.

If $f(x) = 0$, then the equation becomes a homogeneous second order differential equation.
If $f(x) \ne 0$, then the equation becomes a non-homogeneous second order differential equation.

Second-order differential equations have a wide range of applications.

Systems of Electrical Circuits

Electrical systems, also called circuits or networks, are designed as combinations of three components: resistor $\left( {\rm{R}} \right)$, capacitor $\left( {\rm{C}} \right)$, and inductor $\left( {\rm{L}} \right)$. They are defined by resistance, capacitance, and inductance and is generally considered lumped-parameter properties.

The second-order differential equation has derivatives equal to the number of elements storing energy. This differential equation is considered an ordinary differential equation.

Simple Harmonic Motion

The differential equation for the simple harmonic function is given by

$\frac{{{d^2}x}}{{d{t^2}}} = – {\omega ^2}x$, where $\omega $ is the angular velocity of the particle and $T = \frac{{2\pi }}{\omega }$ is the period of motion.

From this, we can conclude that for the larger mass, the period is longer, and for the stronger spring, the period is shorter.

Applications of Partial Differential Equations

The applications of partial differential equations are as follows:

What is PDE?

A Partial differential equation (or PDE) relates the partial derivatives of an unknown multivariable function. Such a multivariable function can consist of several dependent and independent variables.

Laplace Equation: ${\Delta ^2}\phi = \frac{{{\partial ^2}\phi }}{{{\partial ^2}x}} + \frac{{{\partial ^2}\phi }}{{{\partial ^2}y}} = 0$

Heat Conduction Equation: $\frac{{\partial T}}{{\partial t}} = C\frac{{{\partial ^2}T}}{{\partial {x^2}}}$

In physical problems:

To calculate the boundary conditions
To solve boundary value problems using the method of separation of variables

Several problems in Engineering give rise to some well-known partial differential equations. Few of them are listed below.

Wave Equation

$\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {c^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}$

2. One dimensional heat flow equation

$\frac{{\partial u}}{{\partial t}} = {c^2}\frac{{{\partial ^2}T}}{{\partial {x^2}}}$

3. Two dimensional heat flow equation which is steady state becomes the two dimensional Laplace’s equation

$\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = 0$

4. Laplace’s equation in three dimensions

${\Delta ^2}\phi = \frac{{{\partial ^2}\phi }}{{{\partial ^2}x}} + \frac{{{\partial ^2}\phi }}{{{\partial ^2}y}} + \frac{{{\partial ^2}\phi }}{{{\partial ^2}z}} = 0$

Solved Examples

Q.1. The population of a country is known to increase at a rate proportional to the number of people presently living there. If after two years the population has doubled, and after three years the population is $20,000$, estimate the number of people currently living in the country. Ans: Let $N$ denote the number of people living in the country at any time $t$, and let ${N_0}$ denote the number of people initially living in the country. $\frac{{dN}}{{dt}}$, the time rate of change of population is proportional to the present population. Then $\frac{{dN}}{{dt}} = kN$, or $\frac{{dN}}{{dt}} – kN = 0$, where $k$ is the constant of proportionality. $\frac{{dN}}{{dt}} – kN = 0$ which has the solution $N = c{e^{kt}}…….(i)$ At $t = 0,\,N = {N_0}$ Hence, it follows from $(i)$ that $N = c{e^{k0}}$ $ \Rightarrow {N_0} = c{e^{k0}}$ $\therefore \,{N_0} = c$ Thus, $N = {N_0}{e^{kt}}\,………(ii)$ At $t = 2,\,N = 2{N_0}$ [After two years the population has doubled] Substituting these values into $(ii)$, We have $2{N_0} = {N_0}{e^{kt}}$ from which $k = \frac{1}{2}\ln 2$ Substituting these values into $(i)$ gives $N = {N_0}{e^{\frac{t}{2}(\ln 2)}}\,…….(iii)$ At $t = 3,\,N = 20000$. Substituting these values into $(iii)$, we obtain $20000 = {N_0}{e^{\frac{3}{2}(\ln 2)}}$ ${N_0} = \frac{{20000}}{{2\sqrt 2 }} \approx 7071$ Hence, $7071$ people initially living in the country.

Q.2. A metal bar at a temperature of ${100^{\rm{o}}}F$ is placed in a room at a constant temperature of ${0^{\rm{o}}}F$. If, after $20$ minutes, the temperature is ${50^{\rm{o}}}F$, find the time to reach a temperature of ${25^{\rm{o}}}F$. Ans: Newton’s law of cooling is $\frac{{dT}}{{dt}} = – k\left( {T – {T_m}} \right)$ $ \Rightarrow \frac{{dT}}{{dt}} + kT = k{T_m}$ $ \Rightarrow \frac{{dT}}{{dt}} + kT = 0\,\,\left( {\therefore \,{T_m} = 0} \right)$ Which has the solution $T = c{e^{ – kt}}\,…….(i)$ Since $T = 100$ at $t = 0$ $\therefore \,100 = c{e^{ – k0}}$ or $100 = c$ Substituting these values into $(i)$ we obtain $T = 100{e^{ – kt}}\,……..(ii)$ At $t = 20$, we are given that $T = 50$; hence, from $(ii)$, $50 = 100{e^{ – kt}}$ from which $k = – \frac{1}{{20}}\ln \frac{{50}}{{100}}$ Substituting this value into $(ii)$, we obtain the temperature of the bar at any time $t$ as $T = 100{e^{\left( {\frac{1}{{20}}\ln \frac{1}{2}} \right)t}}\,………(iii)$ When $T = 25$ $25 = 100{e^{\left( {\frac{1}{{20}}\ln \frac{1}{2}} \right)t}}$ $ \Rightarrow t = 39.6$ minutes Hence, the bar will take $39.6$ minutes to reach a temperature of ${25^{\rm{o}}}F$.

Q.3. A tank initially holds $100\,l$ of a brine solution containing $20\,lb$ of salt. At $t = 0$, fresh water is poured into the tank at the rate of ${\rm{5 lit}}{\rm{./min}}$, while the well stirred mixture leaves the tank at the same rate. Find amount of salt in the tank at any time $t$. Ans: Here, ${V_0} = 100,\,a = 20,\,b = 0$, and $e = f = 5$, Now, from equation $\frac{{dQ}}{{dt}} + f\left( {\frac{Q}{{\left( {{V_0} + et – ft} \right)}}} \right) = be$, we get $\frac{{dQ}}{{dt}} + \left( {\frac{1}{{20}}} \right)Q = 0$ The solution of this linear equation is $Q = c{e^{\frac{{ – t}}{{20}}}}\,………(i)$ At $t = 0$ we are given that $Q = a = 20$ Substituting these values into $(i)$, we find that $c = 20$ so that $(i)$ can be rewritten as $Q = 20{e^{\frac{{ – t}}{{20}}}}$ Note that as $t \to \infty ,\,Q \to 0$ as it should since only freshwater is added.

Q.4. Find the equation of the curve for which the Cartesian subtangent varies as the reciprocal of the square of the abscissa. Ans: Let $P(x,\,y)$ be any point on the curve, according to the question Subtangent $ \propto \frac{1}{{{x^2}}}$ or $y\frac{{dx}}{{dy}} = \frac{k}{{{x^2}}}$ Where $k$ is constant of proportionality or $\frac{{kdy}}{y} = {x^2}dx$ Integrating, we get $k\ln y = \frac{{{x^3}}}{3} + \ln c$ Or $\ln \frac{{{y^k}}}{c} = \frac{{{x^3}}}{3}$ ${y^k} = {c^{\frac{{{x^3}}}{3}}}$ which is the required equation.

Q.5. Solve the equation $\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}$ with boundary conditions $u(x,\,0) = 3\sin \,n\pi x,\,u(0,\,t) = 0$ and $u(1,\,t) = 0$ where $0 < x < 1,\,t > 0$. Ans: The solution of differential equation $\frac{{\partial u}}{{\partial t}} = \frac{{{\partial ^2}u}}{{\partial {x^2}}}\,……..(i)$ is $u(x,\,t) = \left( {{c_1}\,\cos \,px + {c_2}\,\sin \,px} \right){e^{ – {p^2}t}}\,……..(ii)$ When $x = 0,\,u(0,\,t) = {c_1}{e^{ – {p^2}t}} = 0$ i.e., ${c_1} = 0$. Therefore $(ii)$ becomes $u(x,\,t) = {c_2}\,\sin \,px{e^{ – {p^2}t}}\,……….(iii)$ When $x = 1,\,u(1,\,t) = {c_2}\,\sin \,p \cdot {e^{ – {p^2}t}} = 0$ or $\sin \,p = 0$ i.e., $p = n\pi $. Therefore, $(iii)$ reduces to $u(x,\,t) = {b_n}{e^{ – {{(n\pi )}^2}t}}\sin \,n\pi x$ where ${b_n} = {c_2}$ Thus the general solution of $(i)$ is $u(x,\,t) = \sum {{b_n}} {e^{ – {{(n\pi )}^2}t}}\sin \,n\pi x\,……….(iv)$ When $t = 0,\,3\,\sin \,n\pi x = u(0,\,t) = \sum\limits_{n = 1}^\infty {{b_n}} {e^{ – {{(n\pi )}^2}t}}\sin \,n\pi x$ Comparing both sides, ${b_n} = 3$ Hence from $(iv)$, the desired solution is $u(x,\,t) = 3\sum\limits_{n = 1}^\infty {{b_n}} {e^{ – {{(n\pi )}^2}t}}\sin \,n\pi x$

Learn About Methods of Solving Differential Equations

An equation that involves independent variables, dependent variables and their differentials is called a differential equation. The first-order differential equation is defined by an equation $\frac{{dy}}{{dx}} = f(x,\,y)$, here $x$ and $y$ are independent and dependent variables respectively. The main applications of first-order differential equations are growth and decay, Newton’s cooling law, dilution problems.

Partial differential equations relate to the different partial derivatives of an unknown multivariable function. Several problems in engineering give rise to partial differential equations like wave equations and the one-dimensional heat flow equation. Similarly, the applications of second-order DE are simple harmonic motion and systems of electrical circuits.

Frequently Asked Questions (FAQs)

Q.1. What is a differential equation and its application? Ans: An equation that has independent variables, dependent variables and their differentials is called a differential equation. They are used to calculate the movement of an item like a pendulum, movement of electricity and represent thermodynamics concepts. Graphical representations of the development of diseases are another common way to use differential equations in medical uses.

Q.2. What are the applications of differential equations in engineering? Ans: It has vast applications in fields such as engineering, medical science, economics, chemistry etc. Application of differential equations in engineering are modelling of the variation of a physical quantity, such as pressure, temperature, velocity, displacement, strain, stress, voltage, current, or concentration of a pollutant, with the change of time or location, or both would result in differential equations.

Q.3. What are the applications of differential equations? Ans: Differential equations have many applications, such as geometrical application, physical application. In geometrical applications, we can find the slope of a tangent, equation of tangent and normal, length of tangent and normal, and length of sub-tangent and sub-normal.

Q.4. How many types of differential equations are there? Ans: There are 6 types of differential equations. They are as follows:

Linear Differential Equations
Non-linear Differential Equations
Homogeneous Differential Equations
Non-homogeneous Differential Equations

Q.5. What are the applications of differentiation in economics? Ans: The application of differential equations in economics is optimizing economic functions. For example, the use of the derivatives is helpful to compute the level of output at which the total revenue is the highest, the profit is the highest and (or) the lowest, marginal costs and average costs are the smallest.

Ways To Improve Learning Outcomes: With the development of technology, students may now rely on strategies to enhance learning outcomes. No matter how knowledgeable a...

Visual Learning Style: We as humans possess the power to remember those which we have caught visually in our memory and that too for a...

NCERT Books for Class 6 Social Science 2024: Many state education boards, including the CBSE, prescribe the NCRET curriculum for classes 1 to 12. Thus,...

CBSE Syllabus for Class 9 Social Science: The Central Board of Secondary Education releases the revised CBSE Class 9 Social Science syllabus. The syllabus is...

CBSE Syllabus for Class 8 Maths 2023-24: Students in CBSE Class 8 need to be thorough with their syllabus so that they can prepare for the...

NCERT Books for Class 6 Maths 2025: The National Council of Educational Research and Training (NCERT) textbooks are the prescribed set of books for schools...

CBSE Class 10 Study Timetable: The CBSE Class 10 is the board-level exam, and the Class 10th students will appear for the board examinations for...

CBSE Class 10 Syllabus 2025: The Central Board of Secondary Education (CBSE) conducts the Class 10 exams every year. Students in the CBSE 10th Class...

CBSE Syllabus for Class 11 2025: The Central Board of Secondary Education (CBSE) has published the Class 11 syllabus for all streams on its official...

NCERT Solutions for Class 7 Science Chapter 16 Water – A Precious Resource: In this chapter, students will study about the importance of water. There are three...

NCERT Solutions for Class 7 Science Chapter 10 2024: Respiration in Organisms

NCERT Solutions for Class 7 Science Chapter 10 Respiration in Organisms: NCERT solutions are great study resources that help students solve all the questions associated...

Factors Affecting Respiration: Definition, Diagrams with Examples

In plants, respiration can be regarded as the reversal of the photosynthetic process. Like photosynthesis, respiration involves gas exchange with the environment. Unlike photosynthesis, respiration...

NCERT Solutions for Class 7 Science Chapter 12

NCERT Solutions for Class 7 Science Chapter 12 Reproduction in Plants: The chapter 'Reproduction' in Class 7 Science discusses the different modes of reproduction in...

NCERT Solutions for Class 7 Science Chapter 11

NCERT Solutions for Class 7 Science Chapter 11: Chapter 11 of Class 7 Science deals with Transportation in Animals and Plants. Students need to ensure...

NCERT Solutions for Class 7 Science Chapter 15: Light

NCERT Solutions for Class 7 Science Chapter 15: The NCERT Class 7 Science Chapter 15 is Light. It is one of the most basic concepts. Students...

NCERT Solutions for Class 7 Science Chapter 13

NCERT Solutions for Class 7 Science Chapter 13: Chapter 13 in class 7 Science is Motion and Time. The chapter concepts have a profound impact...

NCERT Solutions for Class 7 Science Chapter 14: Electric Current and its Effects

NCERT Solutions for Class 7 Science Chapter 14: One of the most important chapters in CBSE Class 7 is Electric Current and its Effects. Using...

General Terms Related to Spherical Mirrors

General terms related to spherical mirrors: A mirror with the shape of a portion cut out of a spherical surface or substance is known as a...

Animal Cell: Definition, Diagram, Types of Animal Cells

Animal Cell: An animal cell is a eukaryotic cell with membrane-bound cell organelles without a cell wall. We all know that the cell is the fundamental...

NCERT Solutions for Class 10 Science 2024 – Download PDF

NCERT Solutions for Class 10 Science: The National Council of Educational Research and Training (NCERT) publishes NCERT Solutions for Class 10 Science as a comprehensive...

NCERT Books for Class 12 Chemistry 2024: Download PDF

NCERT Books for class 12 Chemistry: NCERT publishes chemistry class 12 books every year. The NCERT chemistry class 12 books are essential study material for...

CBSE Class 9 Mock Tests 2025: Attempt Online Mock Test Series (Subject-wise)

We all have heard at least once that the secret to success is practice. Some of you could say it's a cliché, but those who...

NCERT Books for Class 10 Maths 2025: Download Latest PDF

NCERT Books for Class 10 Maths: The NCERT Class 10 Maths Book is a comprehensive study resource for students preparing for their Class 10 board exams....

Arc of a Circle: Definition, Properties, and Examples

Arc of a circle: A circle is the set of all points in the plane that are a fixed distance called the radius from a fixed point...

CBSE Class 10 Mock Test 2025: Practice Latest Test Series

CBSE Class 10 Mock Test 2025: Students' stress is real due to the mounting pressure of scoring good marks and getting into a renowned college....

NCERT Solutions for Class 10 2024: Science and Maths

NCERT Solutions for Class 10 2024: Students appearing for the CBSE Class 10 board exam must go through NCERT Solutions to prepare for the exams...

NCERT Solutions for Class 12 2024 – Physics, Chemistry, Maths, Biology

NCERT Solutions for Class 12 2023-24: The NCERT Solutions for Class 12 are meant to help students understand what the subject holds. These solutions are...

39 Insightful Publications

Embibe Is A Global Innovator

Innovator Of The Year Education Forever

Interpretable And Explainable AI

Revolutionizing Education Forever

Best AI Platform For Education

Enabling Teachers Everywhere

Decoding Performance

Leading AI Powered Learning Solution Provider

Auto Generation Of Tests

Disrupting Education In India

Problem Sequencing Using DKT

Help Students Ace India's Toughest Exams

Best Education AI Platform

Unlocking AI Through Saas

Fixing Student’s Behaviour With Data Analytics

Leveraging Intelligence To Deliver Results

Brave New World Of Applied AI

You Can Score Higher

Harnessing AI In Education

Personalized Ed-tech With AI

Exciting AI Platform, Personalizing Education

Disruptor Award For Maximum Business Impact

Top 20 AI Influencers In India

Proud Owner Of 9 Patents

Innovation in AR/VR/MR

Best Animated Frames Award 2024

Take Unlimited Mock Tests related to Your Book Topic

Enter mobile number.

By signing up, you agree to our Privacy Policy and Terms & Conditions

Math Article

Differential Equations Applications

Differential Equation applications have significance in both academic and real life. An equation denotes the relation between two quantity or two functions or two variables or set of variables or between two functions. Differential equation denotes the relationship between a function and its derivatives, with some set of formulas. There are many examples, which signifies the use of these equations.

The functions are the one which denotes some sort of operation performed and the rate of change during the performance is the derivative of that operation, and the relation between them is the differential equation. These equations are represented in the form of order of the degree, such as first order , second order, etc. Its applications are common to find in the field of engineering, physics etc.

In this article, we will learn about various applications in real life and in mathematics along with its definition and its types.

Differential Equations

In terms of mathematics, we say that the differential equation is the relationship that involves the derivative of a function or a dependent variable with respect to an independent variable. It is represented as;

Where x is the independent variable

And y is the dependent variable, as its function is dependent on the values of x.

Y’ denotes one derivative. Similarly, y’’, y’’’, …, and so on, denoted the number of derivatives for all values of x.

There are many applications of differential equations in mathematics based on these formulas.

Types of differential equations

Basically, there are two types of differential equations;

Ordinary Differential Equation(ODE)

Ordinary differential equation involves a relation between one real variable which is independent say x and one dependent variable say y and sum of derivatives y’, y’’, y’’’… with respect to the value of x.

The highest derivative which occurs in the equation is the order of ordinary differential equation . ODE for nth order can be written as;

F(x,y,y’,….,y n ) = 0

Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.

For example, according to the Newton law of cooling, the change in temperature is directly proportional to the difference between the temperature of the hot object or body and temperature of the atmosphere. Therefore, in terms of the differential equation we can represent it as;

Where k is the proportional constant and T is the temperature of the object and T a is the temperature of the atmosphere

Partial Differential Equation(PDE)

In the partial differential equation, unlike ordinary differential equation, there is more than one independent variable.

For example:

$\begin{array}{l}\frac{dz}{dx} + \frac{dz}{dy} = 2z \text{ is a partial differential equations of one order.}\end{array} $
$\begin{array}{l}\frac{d^{2}u}{dx_{2}} + \frac{d^{2}u}{dy^{2}} + 2x + 2y – z \text{ is a partial differential equation of second order.} \end{array} $

Applications of Differential Equations

We can describe the differential equations applications in real life in terms of:

Exponential Growth

For exponential growth, we use the formula;

G(t)= G 0 e kt

Let G 0 is positive and k is constant, then

G(t) increases with time

G 0 is the value when t=0

G is the exponential growth model.

Exponential reduction or decay

R(t) = R 0 e -kt

When R 0 is positive and k is constant, R(t) is decreasing with time,

R is the exponential reduction model

Newton’s law of cooling, Newton’s law of fall of an object, Circuit theory or Resistance and Inductor, RL circuit are also some of the applications of differential equations.

Learn with interactive and interesting learning videos by downloading BYJU’S- The learning App.

MATHS Related Links

Register with BYJU'S & Download Free PDFs

e-Publications@Marquette

< Previous

Home > Dissertations, Theses, and Professional Projects > BACHELOR_ESSAYS > 229

Bachelors’ Theses

The application of differential equations to the laws of physics.

Sherman Baker , Marquette University

Date of Award

Degree type.

Bachelors Essay

Degree Name

Bachelor of Science (BS)

First Advisor

Joseph Wilizewski

In this treatment of the mathematical expression of physical laws, all proofs and expositions of problems that are taken from particular books are acknowledged in the text. Formulas and proofs that are given in a conventional form by authors generally and are the property of no single man are not acknowledged. Such are the alternating current equations, the Mass Action formulas, parts of the thermionic current discussion, and Fourier's equation for the flow of heat.

Some preparation in the sciences on the part of the reader is taken for granted in the presentation of the matter on this thesis. The mathematics involved requires an understanding of linear differential equations and of partial differentiation. Very little knowledge of physics and chemistry is required beyond that taught in college survey courses. The fundamental definitions in the theory of electricity must be known to the reader if he is to follow the first part of the thesis; and to understand the uses of Poisson's equation he must be familiar with the concept of lines of force and with the idea of the electron.

A Thesis Submitted to the Faculty of the College of Liberal Arts of Marquette University in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science

Recommended Citation

Baker, Sherman, "The Application of Differential Equations to the Laws of Physics" (1935). Bachelors’ Theses . 229. https://epublications.marquette.edu/bachelor_essays/229

Since May 03, 2024

Included in

Physics Commons

Advanced Search

Notify me via email or RSS
Collections
Disciplines

Information about e-Pubs@MU

General FAQ

Home | About | FAQ | My Account | Accessibility Statement

Privacy Copyright

IB Maths Resources from Intermathematics

IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides

Differential Equations in Real Life

Real life use of Differential Equations

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time. A differential equation is one which is written in the form dy/dx = ………. Some of these can be solved (to get y = …..) simply by integrating, others require much more complex mathematics.

Population Models One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. The constant r will change depending on the species. Malthus used this law to predict how a species would grow over time.

More complicated differential equations can be used to model the relationship between predators and prey. For example, as predators increase then prey decrease as more get eaten. But then the predators will have less to eat and start to die out, which allows more prey to survive. The interactions between the two populations are connected by differential equations.

The picture above is taken from an online predator-prey simulator . This allows you to change the parameters (such as predator birth rate, predator aggression and predator dependance on its prey). You can then model what happens to the 2 species over time. The graph above shows the predator population in blue and the prey population in red – and is generated when the predator is both very aggressive (it will attack the prey very often) and also is very dependent on the prey (it can’t get food from other sources). As you can see this particular relationship generates a population boom and crash – the predator rapidly eats the prey population, growing rapidly – before it runs out of prey to eat and then it has no other food, thus dying off again.

This graph above shows what happens when you reach an equilibrium point – in this simulation the predators are much less aggressive and it leads to both populations have stable populations.

There are also more complex predator-prey models – like the one shown above for the interaction between moose and wolves. This has more parameters to control. The above graph shows almost-periodic behaviour in the moose population with a largely stable wolf population.

Some other uses of differential equations include:

1) In medicine for modelling cancer growth or the spread of disease 2) In engineering for describing the movement of electricity 3) In chemistry for modelling chemical reactions 4) In economics to find optimum investment strategies 5) In physics to describe the motion of waves, pendulums or chaotic systems.

With such ability to describe the real world, being able to solve differential equations is an important skill for mathematicians. If you want to learn more, you can read about how to solve them here .

If you enjoyed this post, you might also like:

Langton’s Ant – Order out of Chaos How computer simulations can be used to model life.

Does it Pay to be Nice? Game Theory and Evolution . How understanding mathematics helps us understand human behaviour

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here .

Discover more from IB Maths Resources from Intermathematics

Subscribe now to keep reading and get access to the full archive.

Type your email…

Applied Mathematics
Mathematics
Differential Equations

Differential Equations and its Applications

January 2022

Christ University, Bangalore

Discover the world's research

25+ million members
160+ million publication pages
2.3+ billion citations
G F Simmons
Recruit researchers
Join for free
Login Email Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google Welcome back! Please log in. Email · Hint Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google No account? Sign up

Open Access is an initiative that aims to make scientific research freely available to all. To date our community has made over 100 million downloads. It’s based on principles of collaboration, unobstructed discovery, and, most importantly, scientific progression. As PhD students, we found it difficult to access the research we needed, so we decided to create a new Open Access publisher that levels the playing field for scientists across the world. How? By making research easy to access, and puts the academic needs of the researchers before the business interests of publishers.

We are a community of more than 103,000 authors and editors from 3,291 institutions spanning 160 countries, including Nobel Prize winners and some of the world’s most-cited researchers. Publishing on IntechOpen allows authors to earn citations and find new collaborators, meaning more people see your work not only from your own field of study, but from other related fields too.

Brief introduction to this section that descibes Open Access especially from an IntechOpen perspective

Want to get in touch? Contact our London head office or media team here

Our team is growing all the time, so we’re always on the lookout for smart people who want to help us reshape the world of scientific publishing.

Home > Books > Recent Developments in the Solution of Nonlinear Differential Equations

On Some Important Ordinary Differential Equations of Dynamic Economics

Submitted: 04 October 2020 Reviewed: 09 March 2021 Published: 21 April 2021

DOI: 10.5772/intechopen.97130

Cite this chapter

There are two ways to cite this chapter:

From the Edited Volume

Recent Developments in the Solution of Nonlinear Differential Equations

Edited by Bruno Carpentieri

To purchase hard copies of this book, please contact the representative in India: CBS Publishers & Distributors Pvt. Ltd. www.cbspd.com | [email protected]

Chapter metrics overview

1,168 Chapter Downloads

Impact of this chapter

Total Chapter Downloads on intechopen.com

Total Chapter Views on intechopen.com

Mathematical modeling in economics became central to economic theory during the decade of the Second World War. The leading figure in that period was Paul Anthony Samuelson whose 1947 book, Foundations of Economic Analysis, formalized the problem of dynamic analysis in economics. In this brief chapter some seminal applications of differential equations in economic growth, capital and business trade cycles are outlined in deterministic setting. Chaos and bifurcations in economic dynamics are not considered. Explicit analytical solutions are presented only in relatively straightforward cases and in more complicated cases a path to the solution is outlined. Differential equations in modern dynamic economic modeling are extensions and modifications of these classical works. Finally we would like to stress that the differential equations presented in this chapter are of the “stand-alone” type in that they were solely introduced to model economic growth and trade cycles. Partial differential equations such as those which arise in related fields, like Bioeconomics and Differential Games, from optimizing the Hamiltonian of the problem, and stochastic differential equations of Finance and Macroeconomics are not considered here.

Walrassian condition
Marshallian condition
homogeneous function
Cobb–Douglas form
endogenous growth

Author Information

Anastasios tsoularis *.

Leeds Beckett University, Leeds, United Kingdom

*Address all correspondence to: [email protected]

1. Introduction

Ordinary differential equations are ubiquitous in the physical sciences and are fundamental for the understanding of complex engineering systems [ 1 ]. In economics they are used to model for instance, economic growth, gross domestic product, consumption, income and investment whereas in finance stochastic differential equations are indispensable in modeling asset price dynamics and option pricing. The vast majority of the ordinary differential equations in economic are autonomous differential equations or difference equations, where time is an implicit variable, whereas the more difficult to solve delay (differential-difference) equations have received much less attention. Difference equations seem a more natural choice of modeling economic processes as key economic variables are monitored at discrete time units but they can present significant complications in their asymptotic behavior and are thus more difficult to analyse. Differential equations on the other hand, can be more amenable to asymptotic stability analysis. Partial differential equations, usually of the second order, for functions of at least two variables arise naturally in modern macroeconomics from solving an optimization problem formulated in a stochastic setting and in optimal control theory. Two books that are recommended for delving deeper into the- economic applications of differential equations are the introductory one by Gandolfo [ 2 ] and the more advanced by Brock and Malliaris [ 3 ]. Both books are excellent sources for ordinary differential equations in economic dynamics. A more recent book which requires strong mathematical background is by Acemoglu [ 4 ].

2. Some differential equations of neoclassical growth theory and business cycles

Some of the most important differential equations developed by economists during a period spanning over sixty years are presented in this section. Most of them beginning with Solow’s development of a growth model, which was partly motivated by the works of Harrod and Domar, are models from Neoclassical Growth Theory. The main postulate of Neoclassical Growth Theory is that economic growth is driven by three elements: labour, capital, and technology. Economic growth is an important topic in economics and Solow’s growth model is the first topic taught in undergraduate economics because of its underlying simplicity and importance as argued by Acemoglu [ 5 ]. The differential equation by Samuelson is concerned with demand and supply scenarios. Phillips’ work is the earliest attempt to employ classical feedback control theory in order to steer a national economy towards a desired target. The remaining works are differential equations with time lags inherently present in production and capital accumulation. Due to space limitations, the exposition is somewhat uneven with full mathematical analyses of most models and cursory treatments of those with time lags. The choice of the differential equations presented in this chapter is a judicious one, the list is by no means exhaustive, but is meant to afford a glimpse into how the mathematical thinking of some famous economists has influenced the economic growth theory in the twentieth century.

2.1 Harrod-Domar

The Harrod-Domar model was developed independently by Roy Harrod [ 6 ] and Evsey Domar [ 7 ] to analyze business cycles originally but later was used to explain an economy’s growth rate through savings and capital productivity. Output, Y , is a function of capital stock, K , Y = F K , and the marginal productivity, dY dK = c = constant . The model postulates that the output growth rate is given by

where s is the savings rate, and δ the capital depreciation rate. The straightforward solution,

clearly demonstrates that increasing investment through savings and productivity boosts economic growth but does not take into account labour input and population size.

2.2 Samuelson

In his 1941 Paul Samuelson [ 8 ] paper employed simple differential equations to investigate the stability of equilibrium for several demand–supply scenarios. The simplest stability analysis was carried out under the Walrasian and Marshallian assumptions. In the former price increases (decreases) if excess demand is positive (negative), whereas in the latter quantity increases (decreases) if excess demand price is positive (negative). Excess demand is the difference between the quantity that buyers are willing to buy and the quantity that suppliers are willing to supply at the same price. Excess demand price is the difference between the price that buyers are willing to pay for a given quantity and the price required by the suppliers.

Let D p α and S p denote the demand and supply functions of price, p , respectively with α a shift parameter representing “taste”. At equilibrium, price, p ∗ , and quantity, q ∗ , are given by

It is the task of comparative statics to show the determination of the equilibrium values of price and quantity and their sensitivity on the “taste” parameter, α .

The dynamic formulation of the Walrasian assumption is

Retaining the first order term in a Taylor series expansion near the equilibrium, p ∗ , we obtain the following linear differential equation

with solution for an initial price, p 0

The equilibrium is stable if dD dp p ∗ < dS dp p ∗ . Price must rise when demand increases.

The dynamic formulation of the Marshallian assumption is

Neglecting high order terms and using the trivial elementary calculus result, d p D dq = 1 dD dp , d p S dq = 1 dS dp , we obtain

The equilibrium is stable if 1 dD dp q ∗ < 1 dS dp q ∗ . Quantity supplied must rise when demand increases, while the change in price is dependent upon the algebraic sign of the supply curve’s slope.

Production function: = F K L , the quantity of goods by K units of capital and L units of labour at time t . In a closed economy where all output is invested or consumed,

where C t and I t are the consumption and investment functions respectively.

An important assumption of the model are the Inada conditions [ 10 ]

In the limits.

The Inada conditions ensure that F is strictly concave with slope decreasing from infinity to zero.

The function F is linearly homogeneous of degree 1 in K and L (in economic terms this is known as constant returns to scale, increasing capital and labour by a certain amount, results in a proportional rise of production) if

In particular, choosing α = 1 L and set y = Y L , k = K L , representing the output and capital per worker respectively

Growth of Capital in Economy: The growth of the capital stock, K , is equivalent to growth in investment, I , which is used to increase capital subject to depreciation. Depreciation of capital stock will be accounted for so that I is essentially investment = rate of change of capital + capital depreciation rate

where δ is the constant capital depreciation rate.

Letting c t and i t denote the consumption and investment per labour unit

Growth of the Labour Force with full employment: The assumption in the labour market is that the labour supply is equivalent to the population. There is no unemployment and the growth of labour as function of time follows an exponential growth pattern:

The fundamental differential equation of economic growth is then

The differential equations and production functions outlined in these three assumptions are the fundamental elements for Solow’s basic differential equation. In Solow’s paper, a constant fraction of income is allocated to savings, in particular, = y t − c t = f k − 1 − s f k = sf k , so that

The equilibrium solution to the basic differential equation is found from sf k = δ + n k . A well-known function is the Cobb–Douglas production function, Y K L = α K β L 1 − β , 0 < β < 1 , where β is the elasticity of output, K Y ∂ Y ∂ K , with respect to capital. The use of the Cobb–Douglas production function is justified because it exhibits constant returns to scale: If capital and labour are both increased by the same factor, λ > 1 , output will be increased by exactly the same proportion, Y K L = λ α K β L 1 − β . Also the marginal product, ∂ Y ∂ K , ∂ Y ∂ L , diminishes as either K or L increases since ∂ 2 Y ∂ K 2 < 0 , ∂ 2 Y ∂ L 2 < 0 . Introduce k = α K L β = α k β , so the differential equation becomes

From dk dt = 0 , k ∗ = sα δ + n 1 1 − β . Substituting k ∗ = sα δ + n 1 1 − β into y = α k β , the steady state level of per capita income is

The output per unit growth converges to n :

A multiplicative factor in the form of technological progress, t = A 0 e gt , can be introduced in the production function, so that, Y t = aK t β A t L t 1 − β and k t = K t A t L t , leading to

The first order nonlinear differential equation has solution

This solution includes the solution to the labour growth only model, n = 0 . The steady state is

Differentiation of dk dt = sa k β − δ + n + g k with respect to k at k ∗ gives β − 1 δ + n + g < 0 , the equilibrium is stable. The steady state level of per capita income is

a constant, since s , δ , n , g are all constant.

Y t = α K β A 0 L 0 e g 1 − β + n t 1 − β = a k β A 0 L 0 e g 1 − β + n t . The output per unit growth, 1 Y dY dt , converges to g 1 − β + n .

The Solow residual is the part of growth unexplained by changes in capital and labour. For Y t = aK t β A t L t 1 − β

The growth rate per unit output is

A positive Solow residual would indicate a faster output growth than that of capital and labour.

Phelps [ 11 ] used the neoclassical growth model to address the consumption per unit of labour at equilibrium in the so-called “golden rule”. At equilibrium with labour force growth rate, n , only the consumption per unit of labour is

For a maximum consumption per unit of labour

Since d 2 f d k 2 < 0 , the turning point is a maximum given by df dk = n . The “golden rule” concludes that the marginal output per worker must equal the growth rate of the labour force at maximum per capita consumption.

The Ramsey–Cass–Koopmans model, or RCK model, is a neoclassical model of economic growth which differs from Solow’s model in its inclusion of consumption, based primarily on the work of Ramsey [ 12 ], with later significant extensions by Cass [ 13 ] and Koopmans [ 14 ].

A steady state is when c t = f k − δ + n k .

There is a second equation of the RCK model, the social planner‘s problem of maximizing a social welfare function expressed by the integral

where ρ > 0 is the discount rate and u c t is a strictly increasing concave utility function of consumption. The objective is formally stated thus

The Hamiltonian is

where λ is the costate variable (Lagrange multiplier). From

Also for the costate variable

This is a nonlinear differential equation that describes the optimal evolution of consumption, known as the Keynes-Ramsey rule. Along with the differential equation, dk dt = f k − δ + n k − c t , form the RCK dynamical system which does not admit an analytical solution. At equilibrium,

The Jacobian matrix at equilibrium,

has eigenvalues real and opposite in sign as its determinant is − ∂ u ∂ c ∂ 2 u ∂ c 2 ∂ 2 f ∂ k 2 k ∗ < 0 ( f k and u c are both concave), therefore the equilibrium is a saddle point.

The growth in the Solow model is exogenous, the steady state depends on the exogenous parameters, , g , which are due to outside trends. In the absence of A t L t growth cannot be maintained. The marginal product of capital, ∂ Y ∂ K = aβA t 1 − β L K 1 − β = aβA t 1 − β K L 1 − β , is inversely proportional to the capital per labour, K L . In countries with lower capital per labour the marginal product of capital should be higher which is not the case. The disparity could be attributed to the different g values in A t , which is treated as an exogenously given parameter in the Solow model, so an explanation is lacking.

The production function, Y = F K A L offers increasing returns to scale, that is F λK λA λL > λF K A L .

The change in capital is identical to Solow’s model, dK dt = sY − δK , where s is the fraction in savings, δ is the exogenous capital depreciation rate. Labour, L , is also exogenous, dL dt = nL , and is comprises labour involved in research technology, L A , and labour involved in the production of the final goods, L Y , L = L A + L Y .

Technology is exogenous and evolves in time, dA dt = γ L A θ A φ , 0 < θ < 1 , φ < 1 .

As is evident from the three assumptions, Romer’s growth model consists of three sectors: the research sector of ideas, the intermediate goods sector which implements the ideas of the research sector and the final goods sector which produces the final output.

Let g A be the technology growth rate, taken to be constant along the stable path,

In Romer’s model, the output production function is given by

and the capital dynamics is

The respective stable equilibria are

The labour involved in the production of the final goods, L Y , is determined in Romer [ 15 ] by maximizing the net profit for the final goods sector and obtaining the closed form expression for L Y L = r − n r − n + β g A , where r is the interest rate, and all parameters are exogenous except for g A which is derived endogenously.

A nice accessible exposition of both Solow’s and Romer’s growth models is Chu [ 16 ]. Jones [ 17 ] argued that the predicted scale effects of Romer’s theory of growth is inconsistent with the time-series evidence from industrialized economies and that long-term growth depends on exogenous parameters including the rate of population growth.

2.7 Mankiw, Romer and Weil

Mankiw, Romer and Weil [ 18 ] argued that the marginal product of capital, ∂ Y ∂ K , is lower in poorer countries is due to their deficiency in human capital. Human capital is the accumulation of knowledge and skills achieved through training and education, which are essential ingredients in adding economic value. The production function is of the Cobb–Douglas type

where H t is the human capital stock which depreciates at the same rate, δ , as K t . As in Solow’s model, a fraction of the output, sY t , is saved but in this model, it is split between human and capital stock, s = s H + s K . The evolution of the economy is determined by

The equilibrium is

In the steady state,

Introduce the transformations, x 1 = k k ∗ , x 2 = h h ∗ , so that the equilibrium shifts to 1 1 . Then

For small deviations, ξ 1 , ξ 2 , from the equilibrium the linear system

The eigenvalues of the Jacobian matrix,

are given by the roots of the quadratic

From the production function, 1 − α − β > 0 . Since the sum of the eigenvalues is α + β − 2 < 0 , and the product is 1 − α − β > 0 , both roots have negative real parts and the equilibrium point is stable.

Kaldor [ 19 ] presented a model of the trade cycle involving non-linear investment and saving functions that shift over time in response to capital accumulation or decumulation so that the system moves from stable equilibrium to unstable equilibrium to stable equilibrium again. In Kaldor’s model investment, I , and savings, S , functions are non-linear with respect to the level of activity, X , measured in terms of employment.

Kaldor used a differential equation system with general non-linear forms. Net investment, I , and savings, S , are functions of national income, Y , and capital stock, K :

Also growth in capital determines investment is given by

Since income will rise if investment is greater than savings, the dynamics of the national income is captured by the differential equation

For normal income levels,

∂ I ∂ Y > ∂ S ∂ Y .

For extreme income levels, either low or high,

∂ I ∂ Y < ∂ S ∂ Y .

At equilibrium, where dK dt = 0 , income levels are normal.

2.9 Phillips

National governments design their expenditure policies to steer the national economy towards a desired income. The theory of feedback control or servomechanisms provides the mathematical methodology of correcting deviations of the controlled variables from their target values. Feedback policies applied to economic stability were implemented by Phillips [ 20 ].

If Y is national income and D a is the aggregate demand then for some adjustment coefficient, a > 0 ,

A similar differential equation holds for the actual, D g and target government demand, D g ∗ , with b > 0 , namely,

Aggregate and government demand are related by

where m is the private sector’s marginal propensity to spend.

Eliminate D a to obtain

Differentiate the above to obtain

Proportional, D g ∗ = − k P Y , where k P > 0 . This policy does not prevent income reduction and induces oscillations.

Derivative, D g ∗ = − k D dY dt , where k D > 0 . This policy does not prevent income reduction but avoids oscillations.

Integral, D g ∗ = − k I ∫ 0 t Ydt , where k I > 0 . This policy prevents income reduction but can induce unstable movement.

2.10 Kalecki

Kalecki [ 21 ] was the first economist to investigate the relationship between production lags and endogenous business cycles by considering a closed economic system over a short period of time without trend. A t is the gross capital accumulation (unconsumed goods). There is a “gestation period”, θ , for any investment I t . Deliveries L t are equal to investment orders, I t − θ at time, t − θ :

Any orders placed during the “gestation period”, t − θ t , remain unfulfilled, A t is equal to the average of investment orders I t allocated during the period t − θ t :

If K t is the capital stock, and U its physical depreciation

The rate of change in investment is for some constants, m > 0 , n > 0 :

Denoting the deviation of I t from the constant demand for restoration of the depreciated industrial equipment U by J t = I t − U , and differentiating J t

During the interval t ∈ − θ 0 Kalecki assumed that J t = 0 . A standard way to solve this differential equation with delay is to assume a solution of the form, D e αt , with D and α (where α is a complex number), to be determined. The general solution of the differential equation for some constants, c 1 and c 2 is

The sign of the real parameter, b , classifies the behavior of the model as explosive for b > 0 , cyclical for b = 0 , and damped for b < 0 .

2.11 A Solow model with lags

Zak [ 22 ] considered a version of the Solow model with delay. Capital can be used τ periods later, so at time t , the capital to be put into productive use is k t − τ . If f k is the production function, s ∈ 0 1 is the constant savings rate and δ ∈ 0 1 is the constant capital depreciation rate, Zak’s model is

At equilibrium,

Deviations of the form, e t , from equilibrium are governed by

with characteristic equation

In many cases depending on the initial conditions, the roots of the characteristic equation have real parts with opposite signs, indicating the presence of a saddle point unlike Solow’s stable model. The model exhibits endogenous cycles when the roots are purely imaginary.

2.12 Goodwin

Goodwin [ 23 ] presented a nonlinear model of nonlinear business cycles with time lags between decisions to invest and the corresponding outlays. Changes at time, t , in income, y t , induce investment outlays, O i t + θ , at a later time, t + θ . Therefore

Hence the nonlinear delay differential equation modeling the evolution of income is

where O t is autonomous investment outlay and ϵ , α are constants. The derivative, dφ ( y ) ̇ d y ̇ , measures the rate of growth in investment with relative to the income growth, termed as acceleration coefficient. Expanding the two leading terms in Taylor series and neglecting higher order terms, Goodwin obtained the nonlinear delay differential equation

Goodwin assumed that O t is constant, O t = O ∗ , and introduced a new variable

where O ∗ 1 − α is the income at equilibrium. The transformed differential equation is then

The asymptotic behavior of the transformed equilibrium, z = 0 , is determined by the eigenvalue solutions of the characteristic equation

with characteristic roots,

can be either positive or negative, both eigenvalues have positive or negative real parts. So if φ ̇ 0 < 1 − α θ + ϵ the deviations from equilibrium are damped oscillatory motions, but if φ ̇ 0 > 1 − α θ + ϵ the system is unstable and drifts away from the locally linearized region of stability.

2.13 A brief literature survey of current research

We close this chapter by providing a very brief snapshot of the current state of the art in theories of economic growth. Most of the very recent works cited are predominantly mathematical in nature. There is an enormous literature, not touched upon here, which employs Econometrics methods, like for instance panel data regression to estimate economic growth based on explanatory variables such as income, investment, policy indicators, education and others over several decades.

In a short article Zhao [ 24 ] discusses how technology was integrated into economic growth by Romer.

Boyko et [ 25 ] use least squares linear regression to determine the values of the coefficients at which the production functions of Cobb–Douglas in Solow’s growth model provide the best fit for available statistical data. Borges et al. [ 26 ] examine the dynamics of Solow’s economic growth model assuming that the labour force growth rate function is a solution of a delay differential equation thereby avoiding the use of exponential growth, L t = L 0 e nt , often criticized as a rather unrealistic choice. Their approach is motivated by the fact that there are delays in entering and retiring an individual from the labour force, relative to their birth date.

Zhang et al. [ 27 ] base their analysis of how the redistribution of emission quotas would impact short-run equilibrium in a specific market of interest and long-run growth on the Solow growth model with endogenous dynamics and exogenous technological shocks.

Zhang [ 28 ] develops an endogenous growth model based on modifications of both Solow’s model by introducing endogenous knowledge. and Romer’s by allowing knowledge to be gained from learning as well as from research.

The paper by Caraballo et al. [ 29 ] is devoted to analysis of the stability of the economy according to an extended version of Kaldor’s economic growth model. They consider the role of the government’s monetary and fiscal policies and we study whether or not a time delay in implementing and the fiscal policy can affect the economic stability.

Dayal [ 30 ] considers long run historical data and uses difference equation simulation to explore the Solow growth model to assess the growth changes in the recent decade.

Perez-Trujillo et al. [ 31 ] investigate the impact of improvement in accessing innovation and knowledge on economic growth and convergence among countries using an augmented Solow-Swan growth model on data from 138 countries.

Turnovsky [ 32 ] discusses contemporary aspects of stabilization policy in reference to Phillips’ contributions in a lengthy paper of substantial mathematical control theory content.

1. P.A. Samuelson. Foundations of Economic Analysis . Harvard University Press, Cambridge, 1947
2. G. Gandolfo. Economic Dynamics . Springer, 1997
3. W.A. Brock, A.G. Malliaris. Differential equations , Stability and Chaos in Dynamic Economics , North Holland, 1989
4. D.Acemoglu , Introduction to Modern Economic Growth , MIT press, 2009
5. D. Acemoglu D. Economic Growth and Development in the Undergraduate Curriculum, The Journal of Economic Education, 2013;44:2:169-177, https://doi:10.1080/00220485.2013.770344
6. R.F. Harrod, An Essay in Dynamic Theory, The Economic Journal, 49(193) (1939), 14–33
7. E.Domar, Capital Expansion, Rate of Growth, and Employment, Econometrica , 14(2) (1946), 14–33
8. P.A. Samuelson, The stability of equilibrium: Comparative statics and dynamics, Econometrica, 9(2) (1941), 97–120
9. R.M. Solow, A contribution to the theory of economic growth, Quarterly Journal of Economics, 70 (1956), 65–94
10. Ken-Ichi Inada, On a Two-Sector Model of Economic Growth: Comments and a Generalization, The Review of Economic Studies 30 (2) (1963), 119–127
11. E.Phelps, The Golden Rule of Accumulation: A Fable for Growth men, The American Economic Review , 51(4) (1961), 638–643
12. F.Ramsey, A Mathematical Theory of Saving, The Economic Journal, 38(152) (1928), 543–559
13. D.Cass, Optimum growth in an aggregative model of capital accumulation, Review of Economic Studies, 32 (1965), 233–240
14. T. Koopmans, On the concept of optimal economic growth, in (Study Week on the) Econometric Approach to Development Planning, chap. 4, pp. 225–87, North-Holland Publishing Co., Amsterdam, 1965
15. P.M. Romer, Endogenous technological change, Journal of Political Economy, 98(5) (1990), S71-S102
16. A.C. Chu. From Solow to Romer: Teaching endogenous technological change in undergraduate economics, International Review of Economics Education, 2018;27:10-15, https://doi:10-1016/j.iree.2018.01.006
17. C.I.Jones, R&D-Based Models of Economic Growth, Journal of Political Economy, 103(4) (1995), 759–784
18. N.G.Mankiw, D.Romer, D.N.Weil, A Contribution to the Empirics of Economic Growth, The Quarterly Journal of Economics, 107(2) (1992), 407–437
19. N.Kaldor, A model of the trade cycle, The Economic Journal, 50(197) (1940), 78–92
20. A.W. Phillips, Stabilisation policy in a closed economy, The Economic Journal, 64(254) (1954), 290–323
21. M.Kalecki, A Macrodynamic Theory of Business Cycles, Econometrica, 3(3) (1935), 327–344
22. P.J.Zak, Kaleckian Lags in General Equilibrium, Review of Political Economy, 11(3) (1999), 321–330
23. R.M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica, 19(1) (1951), 1–17
24. R. Zhao, Technology and economic growth: From Robert Solow to Paul Romer, Hum Behav & Emerg Tech. 2019;1:62–65, https://doi.org/10.1002/hbe2.116
25. A. A. Boyko, V. V. Kukartsev, V. S. Tynchenko, L. N. Korpacheva, N. N. Dzhioeva, A. V. Rozhkova, S. V. Aponasenko, Using linear regression with the least squares method to determine the parameters of the Solow model, (2020) J. Phys.: Conf. Ser. 1582 012016
26. M.J. Borges, F.Fabião, J.Teixeira, Long Cycles Versus Time Delays in a Modified Solow Growth Model. In: Bilgin M.H., Danis H., Demir E. (eds) Eurasian Economic Perspectives. Eurasian Studies in Business and Economics , 14(1) (2020). Springer, Cham. https://doi.org/10.1007/978-3-030-53536-0_25
27. A.R.Zhang, F.Zandi, H.Kim, A Simple Macroeconomic Model of Decentralized Emission Markets Based on the Solow Growth Model, 05 May 2020, https://doi.org/10.3389/fbloc.2020.00018
28. W-b Zhang, Creativity, Returns to Scale and Growth by Integrating Solow, Dixit-Stiglitz, and Romer, Eastern Journal of Economics and Finance , 5(1) (2020), 1–16
29. Caraballo, T., & Silva, A. (2020). Stability analysis of a delay differential Kaldor’s model with government policies. Mathematica Scandinavica, 126(1), 117–141. https://doi.org/10.7146/math.scand.a-116243
30. V. Dayal (2020) Growth Data and Models. In: Quantitative Economics with R. Springer, Singapore. https://doi.org/10.1007/978-981-15-2035-8_11
31. M. Perez-Trujillo, M. Lacalle-Calderon, The impact of knowledge diffusion on economic growth across countries, World Development , 132 (2020) https://doi.org/10.1016/j.worlddev.2020.104995
32. S.J. Turnovsky, Stabilization theory and policy: 50 years after the Phillips curve, Economica, 78 (2011), 67–78

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Continue reading from the same book

Published: 08 September 2021

By Svetlin G. Georgiev

298 downloads

By Tarig M. Elzaki and Shams E. Ahmed

693 downloads

By Mesliza Mohamed, Gafurjan Ibragimov and Seripah Aw...

485 downloads

IntechOpen Author/Editor? To get your discount, log in .

Discounts available on purchase of multiple copies. View rates

Local taxes (VAT) are calculated in later steps, if applicable.

Support: [email protected]

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

View all journals
Explore content
About the journal
Publish with us
Sign up for alerts
Open access
Published: 15 August 2024

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Thabet Abdeljawad 1 , 2 , 5 ,
Muhammad Sher 3 ,
Kamal Shah 1 , 3 ,
Muhammad Sarwar 1 , 3 ,
Inas Amacha 2 ,
Manar Alqudah 4 &
Asma Al-Jaser 4

Scientific Reports volume 14 , Article number: 18937 ( 2024 ) Cite this article

Metrics details

Computational biology and bioinformatics
Mathematics and computing

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Introduction

Fractional calculus (FC) is an important discipline of mathematics which is mainly investigate non-integer order integrals and derivatives. For the first time in history FC was discussed by two great mathematician Leibnitz and L. Hospital by asking half order derivative of a function 1 . Because of its complexity researchers did not study the subject at that time. With the passage of time when the technology get advancement and new function have been introduced, the subject got proper attention from the mathematician it deserved. Currently, FC has attracted attention of many researchers in the fields of science and engineering. For example author have used FC for the solutions of problem in signal processing 2 , complex systems related to control theory 3 , diffusion problem in physics 4 , problem related economic terms like interest rates, commodity prices and dynamics of market fluctuations 5 and accurate dimensional analysis of image processing 6 . Many applications of FC can also be found in biology, chemistry and other diverse fields of sciences. For instance, authors 6 studied a super-twisting sliding mode control of robotic manipulator using the concepts of FC. Shah, et.al 7 studied an epidemic model of dengue fever disease using fractional non-singular derivative. Ahmad, et.al 8 investigated an adaptive control of a robotic manipulator with a delay in the input by applying the fractional order derivative. Shaikh, et.al 9 conducted a detailed mathematical analysis of HIV/AIDS disease by using non-singular type derivative. In the same way, authors 10 applied FC concepts to study mathematical of COVID-19 using real data of Nigeria. Proceeding with the mentioned process, researchers 11 studied mathematical model for the evolution of COVID-19 outbreak in India.

One of the interesting property of fractional derivative is that it does not have a unique definition. Various definitions, like the Riemann-Liouville (R-L) and the Caputo derivatives 12 , the conformable derivative 13 , the Hilfer derivative 14 , the Harmard derivative 15 and some other are available in literature. These ideas have recently been used as the foundation for breaking down numerous mathematical problems, we refer to 16 , and 17 . Problem related to heterogeneities in materials can not be studied using the above mentioned concepts 18 . Researchers 19 used modified Caputo definition by replacing the singular kernel by non-singular kernel and named the new operator is Caputo-Fabrizio derivative(CFD). Authors 18 studied some important properties of the aforesaid operators. Many problems have been analyzed by using mentioned derivative for existence theory and introduced various properties for the mentioned operator. For some recent work on the mentioned topic, we refer 20 , and 21 and references therein. In CFD researchers identified property of locality in the kernel used in the operator. To overcome this difficulty, Atangana and Baleanu 22 in 2016 replaced the kernel of exponential by Mittag-Leffler in the CFD. The newly defined definition was called ABC derivative and applied by different authors for the solution of real world problem. For instance, Xu et al. 23 discussed HIV-1 model, authors 24 explained in details the Klein-Gordan equation using ABC differential operators. Using ABC derivative with fractional order, authors 25 computed the numerical results for parabolic partial differential equations. In additional, researchers 26 applied ZZ Transform to compute the approximate solution of Fokker Plank equations using fuzzy concepts with ABC derivative.

Furthermore, fractional differential operators, as previously stated, are insufficient to adequately explain problems involving irregular forms and geometry. To describe these kinds of behaviours, the fractal notion has been introduced in the literature. Historically, fractal was mainly used from the $17^{th}$ century and there is a disagreement among mathematicians on how to define fractals formally. Weierstrass in 1872 presented a function with graph which nowadays called fractal. Different mathematician define fractal in different ways like a fractal is a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. In other words, fractal is a fragmented geometrical shapes that may be split into figments, each figment represent a copy of the whole shape. Because they appear in the geometric representations of the majority of chaotic processes, fractals have thus been noted by authors as being significant to the theory of chaos. Atangana 27 has made a significant contribution to this field by bridging the gap between FC and fractal calculus. The aforementioned scholar established numerous definitions that are now widely used and made a substantial contribution to the subject of fractals fractional theory. Newly proposed operators can help to explain fractal behaviour of complicated physical systems and nonlocal phenomena. For instant, Hu and He 28 explained fractals space time and dimensions. In addition, authors 29 . Furthermore, Qureshi and Atangana 30 have studied diarrheal illness models using fractal operators. In the same way, researchers 31 studied mathematical model of Ebola virus disease using the the mentioned operator. The use of fractal theory to issues including fractal heat exchangers, heat transport in porous media, etc., is growing. Here is where some outstanding accomplishments related to the properties and uses of fractals in porous media should be mentioned. The authors investigated a fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media 32 . An analytical model for the transverse permeability of the gas diffusion layer in proton exchange membrane fuel cells with electrical double layer effects was carried out by Liang et al. 33 . Yu, et.al 34 investigated the Characterization of the behaviour of water migration during spontaneous imbibition in coal by using fractals fractional model. Recently, Ahmad, et.al 35 studied a malaria disease mathematical model using fractals fractional concepts. Author 36 investigated a dynamical problem by using generalized fractal-fractional derivative with Mittag-Leffler kernel. Recently, authors 37 have used fractals-fractional derivative to study re-infection model of COVID-19. Author 38 studied a class of hybrid problems by using confirmable fractal fractional derivative. Researchers 39 have studied a mathematical model by using ABC fractals fractional derivative.

Delay differential equations (DDEs) are a unique class of differential equations where the unknown function depends on both its past and present times. The physical phenomenon has a time delay as a result of these issues. DDEs have been used in a number of disciplines, most notably computational physics and electrodynamics. It has also been successfully used to challenges in the domains of economics, chemistry, engineering, and infectious diseases (delay can be added for the time a disease takes to display its symptoms). It is very helpful for modelling issues with time delay systems or memory effects. Examples include population dynamics with time delays in birth or death rates, control systems with communication delays, and chemical reactions with delayed consequences. Many scholars have devoted a tremendous amount of time to finding solutions to delay-type problems. Authors 40 explained how DDEs are using in mathematical modeling of life sciences phenomenon. Author 41 gave a comprehensive analysis of DDEs. The qualitative analysis of DDEs related to the existence theory of solutions and their properties, we refer to 42 . Recently, authors 43 have published a detailed work on computations of numerical results for fractals fractional DDEs.

Investigating the existence literature, we found that very rare work has been done for the class of hybrid fractals HFDEs. Recently, Shafiullah, et.al 44 deduced some computational and theoretical results for a class of fractals HFDEs involving power law kernel by using fixed point analysis and numerical algorithm. But the mentioned area still has not explored very well under various fractals fractional differential operators. Therefore, it was needed to investigate a class of fractals HFDEs under the Mittag-Leffler type kernel. Hence, motivated by the above discussion, in this study a class of fractal HFDEs of linear perturbation type is selected for qualitative analysis and approximate solution. Here we remark that it is possible to model and describe non-homogeneous physical processes that occur in their form using HFDEs. Because the mentioned area incorporate different dynamical systems as specific examples. This category of differential equations comprises the hybrid derivative of an unknown function that depends on nonlinearity. Here a continuous function for fractals fractional derivatives and a piecewise continuous or discrete function needs for fractals fractional integration and such combination make an hybrid problem. For some numerous applications of the said area, we refer to 45 .

To the best of our knowledge no one has considered the general problem defined as follow:

where $0<\sigma \le 1$ , $0<\lambda <1$ , $\textbf{f},\textbf{g}\in C[I\times R,R]$ . Proportional delay term is shown by the symbol $\lambda$ . Pantograph equations are differential equations that have proportional delay terms. Hence, special kind of functional differential equation with proportional delay is known as the pantograph equation. It appears in a variety of pure and practical mathematics domains, including probability, number theory, quantum physics, electrodynamics, and control systems. Certain kinds of the said equations are using in modeling various process of astrophysic. In the concerned problem ( 1 ), if $\lambda =1$ , the said equation will reduce to usual fractals HFDE. If $\lambda >1$ , the said problem ( 1 ) become illposed. The detail theory about the pantograph equations and various applications were discussed in 46 . Here $\xi$ stands for fractal dimension. Additionally, let $\xi =1$ then, the fractional differential operator that is typically used becomes the considered operator. In past, hybrid differential equations have been studied by using ordinary or usual fractional derivatives. But investigation of aforementioned problems by using fractals fractional derivatives were found very rare in literature. On the other hand, the concerned area has numerous applications in modelling real world problems. Therefore, it was needed to investigate hybrid problems by using the fractals fractional derivatives. Stability theory is an important aspects of the qualitative theory which play important role in constructing various numerical schemes. Researchers have studied various kinds of stabilities for different problems. In this research work, we are interested to investigate the Ulam-Hyers (U-H) stability theory for our considered problem. The aforementioned stability was introduced by Ulam 47 and explained by Hyers 48 . In addition, further the aforesaid stability was utilized for various functional problems by Rassias 49 . Recently, the mentioned stability was investigated for the solution of stochastic 50 , impulsive 51 , 52 , 53 , and piecewise 54 fractional differential problems. By using hybrid fixed point theory, we will establish some important results related to existence theory and stability theory to the mentioned problem. Then by using Lagrange’s interpolation method a general numerical scheme will be developed. At the final stage, an examples from biological sciences is given to elaborate our theoretical and numerical results. Here, we remark that numerical analysis under fractals fractional concepts for hybrid problems has not well studied yet.

Preliminaries

Some basic results are recalled from 12 , 22 , 28 , 30 . Let, $\textbf{J}=[0,\textbf{T}]$ , and $\Omega =C(\textbf{J},R)$ stands for Banach space endowed with norm $\Vert \cdot \Vert$ on $\Omega$ as follow:

Definition 2.1

The R-L fractional integral of order $0<\sigma \le 1$ is defined as follows:

Definition 2.2

ABC type arbitrary order integral with order ${\sigma }>0$ is defined as follow:

such that the right side exists.

Definition 2.3

ABC type arbitrary order derivative of order ${\sigma } (0<{\sigma }\le 1)$ is defined as follow:

The functions $M({\sigma })$ in the definition is called normalization obeys $M(0)=M(1)=1,$ and $E_{{\sigma }}$ is Mittag-Leffler function.

Definition 2.4

Fractal type arbitrary order integral of order ${\sigma }>0$ is defined as follow:

Definition 2.5

Fractal fractional ABC integral with fractals order $\xi \in (0, 1],$ and fractional order $\sigma \in (0, 1]$ is defined as follow:

Definition 2.6

Fractal fractional ABC derivative with fractals order $\xi \in (0, 1],$ and fractional order $\sigma \in (0, 1]$ is defined as follow:

If $x\in L[0, {T}]$ and $x(0)=0$ , then the solution of

is described as follows:

Existence theory

In this part, qualitative theory of problem ( 1 ) will be discussed.

Let $\textbf{f}({\theta },{\varvec{\Phi }(\theta )}),$ and $\textbf{g}({\theta },{\varvec{\Phi }(\lambda \theta )})$ be continuous functions on $\Omega$ , then the solution of fractal-fractional differential equation described by

is given by

Using Lemma 2.7 , we get the solution as follows:

$\square$

We define the operator as follows: $\textbf{Z}:\Omega \rightarrow \Omega$ by:

Divide the above operator Eq. ( 6 ) into two sub operators as follow:

From Eq.( 7 ) and Eq.( 8 ), $\textbf{Z}(\varvec{\Phi })$ can be written as follows:

For additional investigation, the ensuing presumptions are required.

There is a constant $K_1>0,$ such that

One has a constant $K_2>0,$ such that

One has a constant $K_3>0$ , such that

Corresponding to function $\textbf{g}(\theta ,\varvec{\Phi }(\theta )),\ \exists \ K_4>0,$ and $K_5>0$ , which yield

Theorem 3.2

Our suggested problem Eq.( 1 ) has one solution at most under the assumptions $\mathbf {A_1}- \mathbf {A_4}$ , provided that $\textbf{K}=\left( K_1 +K_2 +\frac{K_3 \xi (1-{\sigma })T^{\xi -1}}{M(\sigma )}\right) <1$ .

Here it is needed to prove that $\mathbf {Z_1}$ is a contraction and the operator $\mathbf {Z_2}$ is equi-continuous in order to verify the aforementioned claim. $\textbf{D}=\{\textbf{U}\in {B}:\ \Vert \varvec{\Phi }\Vert \le \sigma \}$ is a bounded, closed, and convex subset of B for this define. It’s obvious that $\mathbf {Z_1}$ is continuous. Assuming that $\varvec{\Phi },\varvec{\bar{\Phi }}\in \textbf{D}$ , taking

As a result, $\mathbf {Z_1}$ is contracted. Taking $\varvec{\Phi }\in \textbf{D}$ , we can now demonstrate that $\mathbf {Z_2}$ is equi-continuous.

As a result, $\mathbf {Z_2}(\varvec{\Phi })$ has a bound. $\mathbf {Z_2}(\varvec{\Phi })$ is continuous, just as $\textbf{f}$ . Additionally, if $\theta _1<\theta _2\in [0, \textbf{T}],$

As a result, $\mathbf {Z_2}$ is bounded and equi-continuous. Since the aforementioned operator is compact, it can be made reasonably compact and consequently entirely continuous by applying the Arzel $\acute{a}$ -Ascoli theorem. Schauder’s fixed point theorem thus guarantees that $\mathbf {Z_2}$ has a minimum of one fixed point. Therefore, there is at least one solution to problem Eq. ( 1 ). $\square$

Theorem 3.3

Assuming $\mathbf {A_1}-\mathbf {A_3},$ there exists a unique solution to our problem if the condition

Take $\varvec{\Phi },\bar{\varvec{\Phi }}\in \textbf{D},$ we have

So, our problem under consideration has a unique solution according to Banach’s contraction principle. $\square$

UH stability

In this section, we will establish the conditions required for U-H type stability. The numerical solution of a problem depends on this kind of stability. What conditions allow the approximation of a problem to be almost equal to the exact answer is the fundamental question about an approximation. In the 1940s, Ulam offered a thorough analysis of this important topic (see 47 ). The stability that was previously discussed was extended and generalized by Hyer and Rassias to become generalized U-H and U-H-Rassias stability (see 48 , 49 ).

Examine the matching perturbation issue of Eq. ( 1 ) as

Here, $\textbf{h}(\theta )\in C([0,T],R),\ni ,$ for $\epsilon >0$ , and $|\textbf{h}(\theta )|\le \epsilon$ . Corresponding to any solution $\varvec{\Phi }$ of problem Eq. ( 10 ), one has

The solution of Eq.( 10 ) is given as follows:

Using Eq. ( 6 ), and Eq. ( 12 ), we have

Theorem 4.1

System Eq. ( 1 ) has a U-H and generalized U-H stable solution if

Given that $\bar{\varvec{\Phi }}\in \Omega$ represents a unique solution to problem Eq. ( 10 ), and ${\varvec{\Phi }}\in \Omega$ is any solution of problem Eq. ( 1 ), then

Thus the solution is U-H stable. Further, let there exists a nondecreasing function say $\varsigma :(0, T)\rightarrow R$ , such that $\varsigma (\epsilon )=\epsilon ,$ then from Eq.( 13 ), we can write

thus, Eq.( 14 ) indicates that the solution is generalized U-H stable. $\square$

Numerical approximation

We will use a numerical approach in this section of the manuscript to solve our considered problem. The Lagrange interpolation method will be used to build the numerical scheme, in accordance with the numerical method 55 . The integral form equivalent to our suggested problem, Eq. ( 1 ), is provided by

Using $\theta =\theta _{n+1}, \quad n=0,1,2,3,\cdots ,$ Eq. ( 15 ) implies

Now approximating function $\textbf{g}$ on $[\theta _{k-1},\theta _{k}]$ with $\textbf{h}=\frac{\theta _{k}-\theta _{k-1}}{n}$ , and using the following interpolations

Demonstration of our analysis

Example 6.1.

To verify our analysis and to demonstrate our numerical scheme, we consider the following problem called Regulation of Haematopoiesis 56 . Haematopoiesis is the biological process by which precursor stem cells reappear and divide to form mature blood cells. The body’s process of producing blood cells is known as hemopoiesis regulation. Red and white blood cells are produced in the bone marrow and subsequently reach the bloodstream. The primary factor in the production of red blood cells are unique hormones found in the kidney called erythropoiesis. If the blood function fails to supply enough oxygen, renal tubular epithelial cells emit about 90% of the erythropoiesis hormones. Because of this problem, the blood’s oxygen content drops and a chemical is released, which causes the bone marrow to produce more blood cells. A message that blood components deliver to the marrow is what causes haematological illness.

Here, we will apply our chosen findings to a popular hematopoiesis biological model known as $``{\varvec{Wazewska-Czyzewska}}\,{\varvec{and}}\, {\varvec{Lasota}}\,{\varvec{Model}}''$ . The concerned model has numerous applications in real world problems, we refer few as 57 , 58 . Time delays have been incorporated into various biological models to illustrate resource regeneration durations, maturation intervals, feeding schedules, reaction times, and so forth. The model in question can be expressed numerically as follows:

In this case, $\varvec{\Phi }(\theta )$ denotes the quantity of red blood cells over time. Red blood cell death rates are indicated by $\theta$ and $\mu$ , while red blood cell formation rates are indicated by p and $\gamma$ , which are positive constants. Furthermore, $\tau$ denotes the amount of time needed to produce one red blood cell.

There are some modifications included in the model mentioned above. Here, we added a linear perturbation term to the left side of problem Eq. ( 18 ) and substituted the proportional delay term for the discrete delay term. The updated model is provided by

From Eq.( 19 ), we have

Now we deduce the assumptions $(\mathbf {A_1}-\mathbf {A_4})$ by performing the given process

Additionally, we compute

Therefore, $\textbf{f},\ \textbf{g},$ and $\textbf{h}$ are functions.fulfil each of the related presumptions. From $(\mathbf {A_1}-\mathbf {A_4}),$ using constants $K_1=\delta ,\ K_2=\kappa ,\ K_3=\gamma -p,\ K_4=\gamma -p$ and $K_5=0$ .

Moreover $M(\sigma )=1-\sigma (1-\frac{1}{\Gamma (\sigma +1)}),\ \delta =0.0020,\ T=10.0,\ \kappa =0.10,\ \mu =0.050,\ p=0.0010,$ and $\gamma =0.250$ , we obtain

In Fig. 1 , we display the geometrical behaviours of $\textbf{K},\ \textbf{K}'$ as:

( a ). Fractional and fractals behaviour of $\textbf{K}$ . ( b ). Fractional and fractals behaviour of $\textbf{K}'$ .

As can be seen in Fig. 1 , both $\textbf{K}<1,\ \textbf{K}'<1$ . Thus, Theorem 3.2 , Theorem 3.3 , and Theorem 4.1 hold in their entirety. Consequently, there is at least one solution for Eq. ( 19 ). The requirement for the solution’s uniqueness is also met. Additionally, for [0, 10], the solution is U-H and generalized U-H stable. To illustrate the dynamic behaviour, we here provide the approximate solutions of Example 6.1 for different fractals fractional orders in Figs. 2 , 3 , 4 , 5 , and 6 , respectively.

Utilizing first set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

Utilizing second set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

Utilizing third set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

Utilizing fourth set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

Utilizing fifth set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

In Figs. 2 , 3 , 4 , 5 , and 6 , we have simulated the numerical results graphically for different fractals fractional order values. We see that in start the density of red-blood cells is increasing exponentially and then start to decrease. The decrease in the density of the said amount of red-blood cells is different. Because under different fractals fractional order the decay processes will be different. Usually the decay process is faster on smaller fractional order and greater fractals order and vice versa. In the same way, the growth phenomenon in dynamical system under fractals fractional order is also affected. The said processes is faster at larger fractional order values. Here, we use the fractional order values in (0, 0.25) and fractals values in (0.90, 1.0) and present the results graphically in Fig. 7 .

Utilizing another set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

Here, we use the fractional order values in (0, 0.1) and fractals values in (0.94, 1.0] and present the results graphically in Fig. 8 .

Utilizing lower values set of different fractional and fractals orders to present the solution graphically of Example 6.1 .

From Figs. 7 and 8 we observe that fractional and fractals orders have significant impact on the dynamical behavior of the problem. At lower fractional order values the growth in population for some time is very fast like exponential growth and then becomes stable.

Here, we remark that as non-local operators of differentiation can incorporate increasingly complicated natural phenomena into mathematical equations. The said area has gotten much attention from number of researchers of nearly every field in the sciences, technology, and engineering. The FC has become interested very well for all researchers of science and technology. Since area devoted to fractals HFDEs has not well investigated using non singular fractals fractional differential operators. Therefore, it was needed to provide a sophisticated analysis and numerical results for young researchers to extend their knowledge in this direction. In this research work we have used ABC fractals fractional derivative. Here, we remark that the said operator has all the characteristics which have by other fractional differential operators. This study presents a theoretical and numerical investigation of a class of fractals HFDEs with ABC fractals fractional derivative. The issue under consideration was a hybrid problem involving linear perturbation. Using some fixed point analysis, sufficient requirements were inferred for the existence and uniqueness of the solution. Since stability theory is an important requirement for approximate solutions to nonlinear problems. Because with the help of stability theory we deduce the stable behaviour of solution and methodology we use. Different concepts in this regards were given in literature for stability analysis. U-H concept is one of the powerful procedure to be used to investigate stability results for different problems. Therefore, U-H stability requirements were developed to solve the aforementioned issue. To interpret the results numerically, a potent interpolation-based numerical technique was developed. Our previously mentioned research was applied to an intriguing example Lasota-Wazewska system in order to illustrate our findings. Every theoretical and numerical result was tested with success. We plan to utilize this research in the future for system of fractals HFDEs using more complex dynamical systems addressing real-world problems.

Data availability

All the data used in this work is included in the paper.

Almeida, R., Malinowska, A. B. & Monteiro, M. T. T. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathe. Methods Appl. Sci. 41 (1), 336–352 (2018).

Article ADS MathSciNet Google Scholar

Das, S. & Pan, I. Fractional order signal processing: introductory concepts and applications (Springer Science & Business Media, 2011).

Google Scholar

Araz, S. İ. Analysis of a Covid-19 model: optimal control, stability and simulations. Alexandria Eng. J. 60 (1), 647–658 (2021).

Article Google Scholar

Awadalla, M. & Yameni, Y. Modeling exponential growth and exponential decay real phenomena by $\psi$ -Caputo fractional derivative. J. Adv. Mathe. Comput. Sci. 28 (2), 1–13 (2018).

Atangana, A. & İğret Araz, S. Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications. Adv. Diff. Equ. 2020 (1), 1–89 (2020).

Article MathSciNet Google Scholar

Ahmed, S., Ahmed, A., Mansoor, I., Junejo, F. & Saeed, A. Output feedback adaptive fractional-order super-twisting sliding mode control of robotic manipulator. Iran. J. Sci. Technol. Trans. Elect. Eng. 45 , 335–347 (2021).

Shah, K., Jarad, F. & Abdeljawad, T. On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alex. Eng. J. 59 (4), 2305–2313 (2020).

Ahmed, S., Wang, H., Aslam, M. S., Ghous, I. & Qaisar, I. Robust adaptive control of robotic manipulator with input time-varying delay. Int. J. Control Autom. Syst. 17 (9), 2193–2202 (2019).

Shaikh, A., Nisar, K. S., Jadhav, V., Elagan, S. K. & Zakarya, M. Dynamical behaviour of HIV/AIDS model using fractional derivative with Mittag-Leffler kernel. Alex. Eng. J. 61 (4), 2601–2610 (2022).

Peter, O. J. et al. Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using atangana-baleanu operator. Comput. Mater. Continua 66 (2), 1823–1848 (2021).

Shaikh, A. S., Shaikh, I. N. & Nisar, K. S. A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control. Adv. Differ. Equ. 2020 (1), 373 (2020).

Article MathSciNet PubMed PubMed Central Google Scholar

Teodoro, G. S., Machado, J. T. & De Oliveira, E. C. A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388 , 195–208 (2019).

Khalil, R., Al Horani, M., Yousef, A. & Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math. 264 , 65–70 (2014).

Hilfer, R. (ed.) Applications of fractional calculus in physics (World scientific, 2000).

Kilbas, A. A. Hadamard-type fractional calculus. J. Korean Math. Soc. 38 (6), 1191–1204 (2001).

MathSciNet Google Scholar

Zhang, T. & Li, Y. Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique. Knowl.-Based Syst. 246 , 108675 (2022).

Khan, M. et al. Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel. PLoS One 18 (3), e0277806 (2023).

Article CAS PubMed PubMed Central Google Scholar

Caputo, M. & Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress Fract. Differ. Appl. 2 (1), 1–11 (2016).

Caputo, M. & Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress Fract. Diff. Appl. 1 (2), 73–85 (2015).

Losada, J. & Nieto, J. J. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1 (2), 87–92 (2015).

Gul, R., Sarwar, M., Shah, K., Abdeljawad, T. & Jarad, F. Qualitative analysis of implicit Dirichlet boundary value problem for Caputo-Fabrizio fractional differential equations. J. Function Spaces 2020 , 1–9 (2020).

Atangana, A. & Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. (2016). arXiv preprint arXiv:1602.03408 .

Xu, C., Liu, Z., Pang, Y., Saifullah, S. & Inc, M. Oscillatory, crossover behavior and chaos analysis of HIV-1 infection model using piece-wise Atangana-Baleanu fractional operator: Real data approach. Chaos, Solitons Fractals 164 , 112662 (2022).

Saifullah, S., Ali, A., Irfan, M. & Shah, K. Time-fractional Klein-Gordon equation with solitary/shock waves solutions. Math. Probl. Eng. 2021 , 1–15 (2021).

Alomari, A. K., Abdeljawad, T., Baleanu, D., Saad, K. M. & Al-Mdallal, Q. M. Numerical solutions of fractional parabolic equations with generalized Mittag-Leffler kernels. Numer. Methods Partial Differ. Equ. 40 (1), e22699 (2024).

Saad Alshehry, A., Imran, M., Shah, R. & Weera, W. Fractional-view analysis of fokker-planck equations by ZZ transform with mittag-leffler kernel. Symmetry 14 (8), 1513 (2022).

Article ADS Google Scholar

Atangana, A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102 , 396–406 (2017).

He, J. H. Fractal calculus and its geometrical explanation. Res. Phys. 10 , 272–276 (2018).

ADS Google Scholar

Hu, Y. & He, J. H. On fractal space-time and fractional calculus. Therm. Sci. 20 (3), 773–777 (2016).

Qureshi, S. & Atangana, A. Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fractals 136 , 109812 (2020).

Srivastava, H. M. & Saad, K. M. Numerical simulation of the fractal-fractional Ebola virus. Fractal Fract. 4 (4), 49 (2020).

Xiao, B., Huang, Q., Chen, H., Chen, X. & Long, G. A fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media. Fractals 29 (01), 2150017 (2021).

Liang, M. et al. An analytical model for the transverse permeability of gas diffusion layer with electrical double layer effects in proton exchange membrane fuel cells. Int. J. Hydrogen Energy 43 (37), 17880–17888 (2018).

Article ADS CAS Google Scholar

Yu, X. et al. Characterization of water migration behavior during spontaneous imbibition in coal: From the perspective of fractal theory and NMR. Fuel 355 , 129499 (2024).

Article CAS Google Scholar

Ahmad, I., Ahmad, N., Shah, K. & Abdeljawad, T. Some appropriate results for the existence theory and numerical solutions of fractals-fractional order malaria disease mathematical model. Res. Control Optim. 14 , 100386 (2024).

Ur Rahman, M. Generalized fractal-fractional order problems under non-singular Mittag-Leffler kernel. Res. Phys. 35 , 105346 (2022).

Eiman, Shah K., Sarwar, M. & Abdeljawad, T. A comprehensive mathematical analysis of fractal-fractional order nonlinear re-infection model. Alex. Eng. J. 103 , 353–365 (2024).

Khan, S. Existence theory and stability analysis to a class of hybrid differential equations using confirmable fractal fractional derivative. J. Frac. Calc. Nonlinear Sys. 5 (1), 1–11 (2024).

El-Dessoky, M. M. & Khan, M. A. Modeling and analysis of an epidemic model with fractal-fractional Atangana-Baleanu derivative. Alex. Eng. J. 61 (1), 729–746 (2022).

Smith, H. An Introduction to Delay Differential Equations with Applications to the Life Sciences 119–130 (Springer, 2011).

Book Google Scholar

Balachandran, B., Kalmár-Nagy, T. & Gilsinn, D. E. Delay differential equations (Springer, 2009).

Balachandran, K., Kiruthika, S. & Trujillo, J. Existence of solutions of nonlinear fractional pantograph equations. Acta Mathe. Sci. 33 (3), 712–720 (2013).

Basim, M., Ahmadian, A., Senu, N. & Ibrahim, Z. B. Numerical simulation of variable-order fractal-fractional delay differential equations with nonsingular derivative. Eng. Sci. Technol. Int. J. 42 , 101412 (2023).

Shafiullah, Shah K., Sarwar, M. & Abdeljawad, T. On theoretical and numerical analysis of fractal-fractional non-linear hybrid differential equations. Nonlinear Eng. 13 (1), 20220372 (2024).

Abbas, M. I. & Ragusa, M. A. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry 13 (2), 264 (2021).

Guo, C., Hu, J., Hao, J., Celikovsky, S. & Hu, X. Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions. Kybernetika 59 (3), 342–364 (2023).

Ulam, S. M. Problems in modern mathematics (Courier Corporation, 2004).

Hyers, D. H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27 (4), 222–224 (1941).

Article ADS MathSciNet CAS PubMed PubMed Central Google Scholar

Rassias, T. M. On the stability of the linear mapping in Banach spaces. Proc. Am. Mathe. Soc. 72 (2), 297–300 (1978).

Rhaima, M., Mchiri, L., Makhlouf, A. B. & Ahmed, H. Ulam type stability for mixed Hadamard and Riemann-Liouville fractional stochastic differential equations. Chaos Solitons Fractals 178 , 114356 (2024).

Huang, J. & Luo, D. Ulam-Hyers stability of fuzzy fractional non-instantaneous impulsive switched differential equations under generalized Hukuhara differentiability. Int. J. Fuzzy Syst. 2024 , 1–12 (2024).

Guo, C., Hu, J., Wu, Y. & Celikovsky, S. Non-Singular Fixed-Time Tracking Control of Uncertain Nonlinear Pure-Feedback Systems With Practical State Constraints. IEEE Trans. Circuits Syst. I Regul. Pap. 70 (9), 3746–3758 (2023).

Peng, Y., Zhao, Y. & Hu, J. On The Role of Community Structure in Evolution of Opinion Formation: A New Bounded Confidence Opinion Dynamics. Inf. Sci. 621 , 672–690 (2023).

Khan, S., Shah, K., Debbouche, A., Zeb, S. & Antonov, V. Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems. Phys. Scr. 99 (2), 025225 (2024).

Khan, M. A. & Atangana, A. Numerical Methods for Fractal-fractional Differential Equations and Engineering: Simulations and Modeling (CRC Press, 2023).

Chen, X., Shi, C. & Wang, D. Dynamic behaviors for a delay Lasota-Wazewska model with feedback control on time scales. Adv. Differ. Equ. 2020 (1), 1–13 (2020).

MathSciNet CAS Google Scholar

Xu, G., Huang, M., Hu, J., Liu, S. & Yang, M. Bisphenol A and its structural analogues exhibit differential potential to induce mitochondrial dysfunction and apoptosis in human granulosa cells. Food Chem. Toxicol. 188 , 114713 (2024).

Article CAS PubMed Google Scholar

Luo, W. et al. Update: Innate lymphoid cells in inflammatory bowel disease. Dig. Dis. Sci. 67 (1), 56–66 (2022).

Article PubMed Google Scholar

Download references

Acknowledgements

The authors are grateful to Prince Sultan University for support via the TAS research lab. Dr. Asma Al-Jaser would like to thank PrincessNourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R406), Princess Nourah bint AbdulrahmanUniversity, Riyadh, Saudi Arabia.

Author information

Authors and affiliations.

Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia

Thabet Abdeljawad, Kamal Shah & Muhammad Sarwar

Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

Thabet Abdeljawad & Inas Amacha

Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan

Muhammad Sher, Kamal Shah & Muhammad Sarwar

Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia

Manar Alqudah & Asma Al-Jaser

Department of Mathematics and Applied Mathematics, School of Science and Technology,Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Thabet Abdeljawad

You can also search for this author in PubMed Google Scholar

Contributions

T.A. edited the work. M.S. has written the paper. K.S. has designed the problem and done numerical. M.S. included theoretical results. I.A. included literature review . M.A. has supervised the work and contributed in the revision. A. A. has edited the final version and done significant work in the revised version.

Corresponding author

Correspondence to Inas Amacha .

Ethics declarations

Competing interests.

The authors declare no competing interests.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ .

Reprints and permissions

About this article

Cite this article.

Abdeljawad, T., Sher, M., Shah, K. et al. Analysis of a class of fractal hybrid fractional differential equation with application to a biological model. Sci Rep 14 , 18937 (2024). https://doi.org/10.1038/s41598-024-67158-8

Download citation

Received : 01 February 2024

Accepted : 08 July 2024

Published : 15 August 2024

DOI : https://doi.org/10.1038/s41598-024-67158-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Fractal Fractional derivative
U-H stability
Numerical results

By submitting a comment you agree to abide by our Terms and Community Guidelines . If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Quick links

Explore articles by subject
Guide to authors
Editorial policies

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Information

Author Services

Initiatives

You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader.

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https://www.mdpi.com/openaccess .

Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.

Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

Original Submission Date Received: .

Active Journals
Find a Journal
Proceedings Series
For Authors
For Reviewers
For Editors
For Librarians
For Publishers
For Societies
For Conference Organizers
Open Access Policy
Institutional Open Access Program
Special Issues Guidelines
Editorial Process
Research and Publication Ethics
Article Processing Charges
Testimonials
Preprints.org
SciProfiles
Encyclopedia

Article Menu

Subscribe SciFeed
Recommended Articles
Google Scholar
on Google Scholar
Table of Contents

Find support for a specific problem in the support section of our website.

Please let us know what you think of our products and services.

Visit our dedicated information section to learn more about MDPI.

JSmol Viewer

Application of the triple laplace transform decomposition method for solving singular (2 1)-dimensional time-fractional coupled korteweg–de vries equations (kdv) +.

Share and Cite

Gadain, H.E.; Bachar, I.; Mesloub, S. Application of the Triple Laplace Transform Decomposition Method for Solving Singular (2 1)-Dimensional Time-Fractional Coupled Korteweg–De Vries Equations (KdV) + . Symmetry 2024 , 16 , 1055. https://doi.org/10.3390/sym16081055

Gadain HE, Bachar I, Mesloub S. Application of the Triple Laplace Transform Decomposition Method for Solving Singular (2 1)-Dimensional Time-Fractional Coupled Korteweg–De Vries Equations (KdV) + . Symmetry . 2024; 16(8):1055. https://doi.org/10.3390/sym16081055

Gadain, Hassan Eltayeb, Imed Bachar, and Said Mesloub. 2024. "Application of the Triple Laplace Transform Decomposition Method for Solving Singular (2 1)-Dimensional Time-Fractional Coupled Korteweg–De Vries Equations (KdV) + " Symmetry 16, no. 8: 1055. https://doi.org/10.3390/sym16081055

Article Metrics

Further information, mdpi initiatives, follow mdpi.

Subscribe to receive issue release notifications and newsletters from MDPI journals

A numerical approach to variational iteration method for system of nonlinear ordinary differential equations

Sinha, Vikash Kumar
Maroju, Prashanth

This paper propose a new application of the variational iteration method (VIM) by using the Adomian polynomial for solving the nonlinear systems of ordinary differential equations. This is basically a generalisation method. Here, we also discuss the convergence of the presented approach in Banach space. Two scientific problems, including the chaotic Genesio system, are considered to verify the efficacy of the present method. The behaviour of the method is observed for various values of x. This is an efficient technique to solve the several application problems. We compare the proposed approach with the existing Adomian's decomposition method. The outcomes show that our suggested strategy is quite simple and effective.

MATHEMATICS

IMAGES

(PDF) Application of Differential Equations using First Order
Differential Equation
Application of differential equations
(PDF) First-order Ordinary Differential Equations and Applications
Differential Equations Summary
Introduction to the differential equations

COMMENTS

Essay about the art and applications of differential equations?
Differential equations is a rather immense subject. In spite of the risk of overwhelming you with the amount of information, I recommend looking in the Princeton Companion to Mathematics, from which the relevant sections are (page numbers are within parts). Section I.3.5.4 for an introductory overview
PDF First-Order Differential Equations and Their Applications
An ordinary differential equation is an equation relating an unknown function of one variable to one or more functions of its derivatives. If the unknown x is a function of t, x x(t), then examples of ordinary differential equations are. dx. = dt t7 cos x, d2x dx. dt2 = x , dt.
1.1: Applications Leading to Differential Equations
1.1: Applications Leading to Differential Equations. In order to apply mathematical methods to a physical or "real life" problem, we must formulate the problem in mathematical terms; that is, we must construct a mathematical model for the problem. Many physical problems concern relationships between changing quantities.
PDF DIFFERENTIAL EQUATIONS: AN APPLIED APPROACH
equations that arise in many, if not most, scientiﬁc and engineering applications. The equations (1.1), (1.2), and (1.3) are examples of a general class of ordinary diﬀerential equations of the form x0 = f(t,x). Here all terms in the equation not involving the derivative have been placed on the right hand side.
7.1: An Introduction to Differential Equations
A differential equation describes the derivative, or derivatives, of a function that is unknown to us. By a solution to a differential equation, we mean simply a function that satisies this description. For instance, the first differential equation we looked at is. ds dt = 4t + 1, d s d t = 4 t + 1,
Differential Equations: Theory and Applications
This is one graduate-level graduate differential equations text that really would support self-study." (William J. Satzer, The Mathematical Association of America, February, 2010) "The book is an introduction to the theory of ordinary differential equations and intended for first- or second-year graduate students. …
Introduction to differential equations
This free OpenLearn course, Introduction to differential equations, is an extract from the Open University module MST125 Essential mathematics 2 . The module builds on mathematical ideas introduced in MST124 Essential mathematics 1, covering a wide range of topics from different areas of mathematics. The practical application to problems ...
500 Examples and Problems of Applied Differential Equations
Authors: Ravi P. Agarwal, Simona Hodis, Donal O'Regan. Highlights an unprecedented number of real-life applications of differential equations and systems. Includes problems in biomathematics, finance, engineering, physics, and even societal ones like rumors and love. Includes selected challenges to motivate further research in this field.
Differential equation
Differential equations came into existence with the invention of calculus by Isaac Newton and Gottfried Leibniz.In Chapter 2 of his 1671 work Methodus fluxionum et Serierum Infinitarum, [2] Newton listed three kinds of differential equations: = = (,) + = In all these cases, y is an unknown function of x (or of x 1 and x 2), and f is a given function. He solves these examples and others using ...
PDF DIFFERENTIAL EQUATIONS FOR ENGINEERS
the relevance of differential equations through their applications in various engineering disciplines. Studies of various types of differential equations are determined by engi-neering applications. Theory and techniques for solving differential equations are then applied to solve practical engineering problems. Detailed step-by-step analysis ...
The Impact of Differential Equations in Our Everyday Lives (9 examples
1- Weather Forecasting. One of the most immediate applications of differential equations that comes to mind is in the field of weather forecasting. Meteorologists rely on mathematical models derived from differential equations to predict the movements and interactions of temperature, pressure, and moisture in our atmosphere.
Application of Differential Equations: Definition, Types, Examples
Applications of Differential Equations: A differential equation, also abbreviated as D.E., is an equation for the unknown functions of one or more variables.It relates the values of the function and its derivatives. Differential equations have applications in various fields of Science like Physics (dynamics, thermodynamics, heat, fluid mechanics, and electromagnetism), Chemistry (rate of ...
Differential Equations Applications
The highest derivative which occurs in the equation is the order of ordinary differential equation.ODE for nth order can be written as; F(x,y,y',….,y n) = 0. Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts.
PDF DIFFERENTIAL EQUATIONS FOR ENGINEERS
Example 5.7 — Single Degree-of-Freedom System. The single degree-of-freedom system described by x(t ), as shown in Figure 5.26(a), is subjected to a sinusoidal load F (t)=F0 sin t. Assume that the mass m, the spring stiffnesses k1 and k2, the damping coefficient c, and F0 and are known.
1.2: Applications Leading to Differential Equations
Such equations are differential equations. They are the subject of this book. Much of calculus is devoted to learning mathematical techniques that are applied in later courses in mathematics and the sciences; you wouldn't have time to learn much calculus if you insisted on seeing a specific application of every topic covered in the course.
The Application of Differential Equations to the Laws of Physics
Some preparation in the sciences on the part of the reader is taken for granted in the presentation of the matter on this thesis. The mathematics involved requires an understanding of linear differential equations and of partial differentiation. Very little knowledge of physics and chemistry is required beyond that taught in college survey courses.
8: Introduction to Differential Equations
We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section. 8.3E: Exercises for Section 8.3
Differential Equations in Real Life
Real life use of Differential Equations Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. They can describe exponential growth and decay, the population growth of species or the change in investment return over time.
(PDF) Differential Equations and its Applications
F or a falling ob ject, a (t) is constant and is equal to g = -9.8 m/s. d h. dt 2=g. integrate both sides of the above equation to obtain: dh. dt =gt +v0. Integrate one more time to obtain: h ( t ...
PDF Numerical Methods for Differential Equations
studied the nature of these equations for hundreds of years and there are many well-developed solution techniques. Often, systems described by differential equations are so complex, or the systems that they describe are so large, that a purely analytical solution to the equations is not tractable. It is in these complex systems where computer ...
On Some Important Ordinary Differential Equations of ...
Mathematical modeling in economics became central to economic theory during the decade of the Second World War. The leading figure in that period was Paul Anthony Samuelson whose 1947 book, Foundations of Economic Analysis, formalized the problem of dynamic analysis in economics. In this brief chapter some seminal applications of differential equations in economic growth, capital and business ...
Analysis of a class of fractal hybrid fractional differential equation
Pantograph equations are differential equations that have proportional delay terms. Hence, special kind of functional differential equation with proportional delay is known as the pantograph equation.
Geometrical Application Of Ordinary Differential Equation
Partial differential equations(PDE) are significantly more difficult than ODE, and we won't talk about it at this moment. Order. Order of a differential equations is the order of the highest derivative in the equation. Order 1: Order 2: Degree. The degree of a differential equation is the degree of the highest derivative in the equation ...
3.5: An Application to Systems of Differential Equations
The simplest differential system is the following single equation: It is easily verified that is one solution; in fact, Equation [eq:diffeq] is simple enough for us to find all solutions. Suppose that is any solution, so that for all . Consider the new function given by . Then the product rule of differentiation gives.
Symmetry
The main aim of this article is to modify the space-time fractionalKdV equations using the Bessel operator. The triple Laplace transform decomposition method (TLTDM) is proposed to find the solution for a time-fractional singular KdV coupled system of equations. Three problems are discussed to check the accuracy and illustrate the effectiveness of this technique. The results imply that our ...
A numerical approach to variational iteration method for system of
This paper propose a new application of the variational iteration method (VIM) by using the Adomian polynomial for solving the nonlinear systems of ordinary differential equations. This is basically a generalisation method. Here, we also discuss the convergence of the presented approach in Banach space. Two scientific problems, including the chaotic Genesio system, are considered to verify the ...

Stack Exchange Network

Essay about the art and applications of differential equations?

2 Answers 2

You must log in to answer this question.

Hot Network Questions

My OpenLearn Profile

About this free course

Introduction to differential equations

500 Examples and Problems of Applied Differential Equations

Department of Mathematics, National University of Ireland, Galway, Ireland

Access this book

Other ways to access

About this book

Similar content being viewed by others

General Systems of Differential Equations

Ordinary differential equations

Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

Table of contents (10 chapters)

Some First-Order Nonlinear Differential Equations

Authors and Affiliations

About the authors

The Impact of Differential Equations in Our Everyday Lives (9 examples!)

What Are Differential Equations?

Differential Equations Applications in Real Life

1- Weather Forecasting

2- Modeling Ecology

3- Modeling Population Dynamics

4- Physics and Engineering

5- Economics and Finance

6- Environmental Science

7- Medicine and Healthcare

8- Modeling Climate Change

9- Applied Engineering Studies

What to read next:

In Conclusion

Recent Posts

Share this article

Table of Contents

Ways To Improve Learning Outcomes: Learn Tips & Tricks

Visual Learning Style for Students: Pros and Cons

NCERT Books for Class 6 Social Science 2024 – Download PDF

CBSE Syllabus for Class 9 Social Science 2023-24: Download PDF

CBSE Syllabus for Class 8 Maths 2024: Download PDF

NCERT Books for Class 6 Maths 2025: Download Latest PDF

CBSE Class 10 Study Timetable 2024 – Best Preparation Strategy

CBSE Class 10 Syllabus 2025 – Download PDF

CBSE Syllabus for Class 11 2025: Download PDF

NCERT Solutions for Class 7 Science Chapter 16 Water – A Precious Resource

Application of Differential Equations: Definition, Types, Examples

Differential Equation: Overview

Applications of Differential Equations in Real Life

Applications of First-order Differential Equations

What is an ODE?

First-Order ODE

Growth and Decay

Newton’s Cooling Law

Applications of Second-order Differential Equations

Second-Order ODE

Systems of Electrical Circuits

Simple Harmonic Motion

Applications of Partial Differential Equations

What is PDE?

Solved Examples

Frequently Asked Questions (FAQs)

Related Articles

NCERT Solutions for Class 7 Science Chapter 10 2024: Respiration in Organisms

Factors Affecting Respiration: Definition, Diagrams with Examples

NCERT Solutions for Class 7 Science Chapter 12

NCERT Solutions for Class 7 Science Chapter 11

NCERT Solutions for Class 7 Science Chapter 15: Light

NCERT Solutions for Class 7 Science Chapter 13

NCERT Solutions for Class 7 Science Chapter 14: Electric Current and its Effects

General Terms Related to Spherical Mirrors

Animal Cell: Definition, Diagram, Types of Animal Cells

NCERT Solutions for Class 10 Science 2024 – Download PDF

NCERT Books for Class 12 Chemistry 2024: Download PDF

CBSE Class 9 Mock Tests 2025: Attempt Online Mock Test Series (Subject-wise)

NCERT Books for Class 10 Maths 2025: Download Latest PDF

Arc of a Circle: Definition, Properties, and Examples

CBSE Class 10 Mock Test 2025: Practice Latest Test Series