A FIRST COURSE OF DIFFERENTIAL
EQUATIONS WITH APPLICATIONS
By
A.H. Siddiqi
And
P. Manchanda
A FIRST COURSE OF DIFFERENTIAL
EQUATIONS WITH APPLICATIONS
Professor A.H. Siddiqi
Department of Mathematical Sciences
King Fahd University of Petroleum & Minerals
Dhahran31261, Saudi Arabia
and
Senior Associate, Abdus Salam International
Center of Theoretical Physics, Trieste, Italy
Dr. P. Manchanda
Department of Mathematics
GuruNanakDevUniversity, Amritsar, Punjab, India
McMillan India, New Delhi, Bangalore, Chennai, Mumbai
To my wife Dr. Azra H. Siddiqi and parents
A.H. Siddiqi
To my husband Dr. K.S. Manchanda and parents
P. Manchanda
CONTENTS
PREFACE
1. Introduction to Differential Equations
1.1 Introduction
1.2 Definitions and Terminology
1.3Initial-Value and Boundary-Value Problems
1.4Differential Equations as Mathematical Models
1.4.1. Population Dynamics (Logistic Model of Population Growth)
1.4.2. Radioactive Decay and Carbon Dating
1.4.3. Supply, Demand and Compounding of Interest
1.4.4 Newton's law of Cooling/Warming
1.4.5 Spread of a Disease
1.4.6 Chemical Reactions
1.4.7 Chemical Mixtures
1.4.8 Draining a Tank.
1.4.9 Series Circuit
1.4.10 Falling Body
1.4.11 Artificial Kidney
1.4.12 Survivability with AIDS( Acquired immunodeficiency )
1.5Exercises
2.First-Order Differential Equations
2.1Separable Variables
2.2Exact Equations
2.2.1Equations Reducible to Exact Form
2.3Linear Equations
2.4Solutions by Substitution
2.4.1Homogenous Equation.
2.4.2Bernoulli’s Equation
2.5Exercises
3.First-Order Differential Equations of Higher Degree.
3.1Equations of the First Order but not of the First Degree
3.2First Order Equations of Higher Degree Solvable for Derivative
3.3Equations Solvable for y
3.4Equations Solvable for x
3.5Equations of the First Degree in x and y –Lagrange’s and Clairaut's Equation
4.Applications of First Order Differential Equations to Real World Systems
4.1Cooling Law
4.2Population Growth and Decay
4.3Radio Active Decay
4.4Mixture of Two Salt Solutions
4.5Series Circuits
4.6Survivability with AIDS
4.7Draining a Tank
4.8Economics and Finance
4.9Mathematical Police Women
4.10Drug Distribution in Human Body
4.11A Pursuit Problem
4.12Harvesting of Renewable Natural Resources
4.13Exercises
5.Higher Order Differential Equations
5.1Initial-Value and Boundary-Value Problems
5.2Homogenous Equations
5.3Non-homogenous Equations
5.4Reduction of Order
5.5Solution of Homogenous Linear Equations with Constant Coefficients
5.6The Method of Undetermined Coefficients
5.7The Method of Variation of Parameters
5.8Cauchy-Euler Equation
5.9Non-linear Differential Equations
5.10Exercises.
6.Power Series Solutions of Linear Differential Equations
6.1Review of Properties of Power Series
6.2Solutions about Ordinary Points
6.3Solutions about Regular Singular Points
6.4Bessel's Equation and Functions
6.5Legendre's Equation and Polynomials
6.6Orthogonality of Functions
6.7Sturm - Liouville Theory
6.8Exercises
7.Modelling and Analysis of Real World Systems by Higher Order Differential Equations
7.1Series Electrical Circuit
7.2Falling Bodies
7.3The Shape of a Hanging Cable - The Power Line Problem
7.4Diabetes and Glucose Tolerance Test
7.5Rocket Motion
7.6Undamped and Damped Motion
7.7Exercises
8.System of Linear Differential Equations with Applications
8.1System of Linear First Order Equations
8.2Matrices and Linear Systems
8.3Homogeneous Systems : Distinct Real Eigenvalues
8.4Homegeneous Systems : Complex and Repeated Real Eigenvalues
8.5Method of Variation of Parameters
8.6Matrix Exponential
8.7Applications
8.7.1 Electrical Circuits
8.7.2Coupled Springs
8.7.3Mixture Problems
8.7.4. Arm Race
8.8Exercises
9.Laplace Transforms and Their Applications to Differential Equations
9.1Definition and Fundamental Properties of the Laplace Transform
9.2The Inverse Laplace Transform
9.3Shifting Theorems and Derivative of Laplace Transform
9.4Transforms of Derivatives and Convolution theorem
9.4.1 The Laplace Transform of Derivatives
9.4.2Convolution
9.4.3Unit Impulses and the Dirac Delta Function
9.5Applications to Differential and Integral Equations
9.6Exercises
10.Numerical Methods for Ordinary Differential Equations.
10.1.Direction Fields
10.2.Euler Methods
10.3.Runga-Kutta Method
10.4.Picard's Method of Successive Approximation
10.5.Exercises
11.Introduction to Partial Differential Equations
11.1Basic Concepts and Definitions
11.2Classification of Partial Differential Equations (PDEs)
11.2.1Initial and Boundary Value Problems
11.2.2Classification of Second-Order Partial Differential Equations
11.3Solutions of First-Order Partial Differential Equations
11.3.1Solutions of Partial Differential Equations of First Order with Constant Coefficients
11.3.2Lagrange Method for Partial Differential Equations with Variable Coefficients
11.3.3Solving Non-Linear Partial Differential Equations of First-Order
11.3.4Charpit’s Method for Solutions of Special Type of First-Order Nonlinear PDEs
11.3.5Geometric Concepts related to Partial Differential Equations of First Order
11.4Solutions of Linear Partial Differential Equations of Second-order with Constant Coefficients
11.4.1Homogeneous Equations
11.4.2Non-homogeneous Equations
11.5Monge’s Method for a Special Class of Nonlinear Partial Differential Equations (Quasi-linear Equations) of the Second-Order
12.Partial Differential Equations of Real World Systems
12.1Partial Differential Equations as Models of Real World Systems
12.2Elements of Trigonometric Fourier Series for Solutions of Partial Differential Equations
12.3Method of Separation of Variables for Solving Partial Differential Equations
12.3.1The Heat Equation
12.3.2The Wave Equation
12.3.3The Laplace Equation
12.4Solutions of Partial Differential Equations with Boundary Conditions
12.4.1The Wavelet Equation with Initial and Boundary Conditions
12.4.2The Heat Equation with Initial and Boundary Conditions
12.4.3The Laplace Equation with Initial and Boundary Conditions
12.4.4Black-Scholes Model of Financial Engineering Mathematics
12.5Exercises
13.Calculus of Variations with Applications.
13.1Variational Problems with Fixed Boundaries
13.2Applications to Concrete problems
13.3Variational Problems with Moving Boundaries
13.4Variational Problems Involving Derivatives of Higher Order and Several Independent Variables
13.5Sufficient Condition for an Extremum – Hamilton-Jacobi Equation
13.6Exercises
Bibliography
Appendices
Appendix – A
Appendix - B
Appendix - C
Solution, Hints and Answer of Selected Exercises
Index.
Preface
The book is designed to present the theme of differential equations at undergraduate level that emphasizes solution techniques and applications. Differential equations are interesting and important because they express relationships involving rates of change. Such relationships form the basis for developing ideas and studying phenomena in various fields of science, economics, engineering, medicine and without any exaggeration every aspect of human knowledge. Sir Isaac Newton encountered concepts related to differential equations in his studies of Physical World. Gottfried Leibnitz has also contributed significantly in formulation of fundamental concepts related to differential equations. Euler, Lagrange and Cauchy provided basic methodology discussed in this book. The material presented in this book provides foundation for scientific, technological and industrial developments.
The essential features of this book are:
- Narrative that is written at a level that can be read and understood by the average differential equations student.
- A large number of applications so that most equation categories have an identifiable application area.
- A large number of graded exercises, including applied problems that can be understood and solved by any one with knowledge of calculus.
- Solutions, Hints and Answers of Selected Exercises.
Differential equations occupy pivotal position in all established disciplines of mathematics such as Pure Mathematics, Applied Mathematics, Industrial Mathematics, Geomathematics, Biomathematics, and Financial Mathematics. New mathematical methods, both analytic as well as numeric, have been developed and currently researchers all over the world are engaged to develop refined methods for solutions of differential equations presented in this book. Higher dimensional analogue of models of this book constitute emerging areas of mathematical research.
There are thirteen chapters in this book. First seven chapters are devoted to ordinary differential equations. Chapter eight deals with the systems of ordinary differential equations. The Laplace transform and its applications to differential equations are discussed in Chapter 9. Numerical Methods for differential equations are given in Chapter 10. Chapters 11 and 12 are devoted to partial differential equations such as Heat equation, the wave equation, the Laplace equation, the Black-Scholes equation, the transport equation, Poisson equation, Helmholtz equation, Telegraph equation, KdV equation, are presented. Chapter 13 deals with the Calculus of Variations with applications. Methods for solutions of ordinary and partial differential equations discussed in the previous chapters are needed to carry out solution of the problems of the calculus of variations.
Basic definitions and terminology along with the description of real world systems and their representation by differential equations of first order are presented in Chapter 1. In Chapter 2, we discuss methods for solving differential equations of first-order belonging to certain classes such as separationof variables, exact, linear, homogeneous and Bernoulli categories of equations. Chapter 3 is devoted to differential equations of first-order and higher degree. Applications of first-order differential equations to 12 areas; namely population dynamics, radio-active decay and carbon dating, supply and demand and compounding of interest, Newton’s law of cooling and warming, spread of diseases, chemical reactions, chemical mixtures, draining of tank, series circuit, falling body, artificial kidney and survivability with AIDS are treated in Chapter 4. Chapter 5 develops method for solving differential equations of higher order related to linear equations with constant applications and Cauchy-Euler equation. Methods include method of undetermined coefficients and the method of variation of parameters. Chapter 6 deals with the power series solution of differential equations and Sturm-Liouville Theory related to eigenvalues problems. Modelling and Analysis of real world systems by higher order differential equations are elaborated in Chapter 7. Modelling of series electrical circuit, falling bodies, the shape of hanging cables, diabetes and glucose tolerance test, rocket motion and undamped and damped motion are considered.
Solution of the system of linear differential equations and their applications are given in Chapter 8. Laplace transforms and their applications to differential equations are included in Chapter 9. Chapter 10 deals with numerical methods of ordinary differential equations such as Euler, Runga-Kutta and Picard. Chapter 11 comprises basic elements of partial differential equations of first-order with constant coefficients, the Lagrange method for solving linear partial differential equations with variable coefficients; Charpit’s method for solving nonlinear partial differential equations of first-order, and methods including Monge’s method for solving partial differential equations of second-order.
Chapter 12 treats partial differential equations representing sixteen real world systems, introduction to Fourier series, method of separation of variables for solving partial differential equations and heat, wave, Laplace and Black-scholes equations with boundary and initial value problems.
Calculus of variations and applications is introduced in Chapter 13. Calculus of variations problem is to determine a function that minimizes or maximizes (extremizes) an integral. Necessary conditions for existence of solutions of the problem of calculus of variations are discussed under different conditions. Euler-Lagrange, Hamlinton-Jacobi, Ostrogradskyequations and their applications to physical phenomena are discussed.
The present book could be an appropriate text book for an undergraduate course for science and engineering students. Even this book could be used by students of industrial mathematics and financial engineering mathematics.
The book is based on the content of courses taught by the authors to science and engineering students in different parts of the world. The first author would like to thank the King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia, where a major part of the book has been used for teaching certain courses.
The authors would also like to thank Mr. Rajiv Beri, General Manager, Macmillan India who invited them to write the book.
A.H. Siddiqi and
P. Manchanda
March 28, 2005
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