Many scientific and real-world problems that occur in science, engineering, and medicine can be represented in differential equations. There is a vital role for differential equations in studying the behavior of different types of real-world problems. Thus, it becomes crucial to know the existence uniqueness properties of differential equations and various methods of finding differential equation solutions in explicit form. It is also essential to know different kinds of differential equations in terms of eigenvalues, termed eigenvalue problems, and some special functions used in finding the solution to differential equations. The study of nonlinear problems also plays a significant role in different real-world situations. There is a necessity to know the behavior of solutions of nonlinear differential equations. Still, there are very few forms of differential equations whose solution can be found in explicit form. For the differential equations whose solutions cannot be found in explicit form, one has to study the properties of solutions of the given differential equation to guess an approximate solution of it. This book aims to introduce all the necessary topics of differential equations in one book so that laymen can easily understand the subject and apply it in their research areas. The novel approach used in this book is that I have introduced different analytical methods for finding the solution of differential equations with sufficient theorems, corollaries, and examples, and the geometrical interpretations in each topic.
This textbook is intended to study the theory and methods of finding the explicit solutions to differential equations, wherever possible, and in the absence of finding explicit solutions. It is intended to study the properties of solutions to the given differential equations. This book is based on syllabi of the theory of differential equations prescribed for postgraduate students of mathematics and applied mathematics in different institutions and universities of India and abroad. This book will be helpful for competitive examinations as well.
Author(s): Ramakanta Meher
Series: River Publishers Series in Mathematical, Statistical and Computational Modelling for Engineering
Publisher: River Publishers
Year: 2023
Language: English
Pages: 291
City: Gistrup
Front Cover
Textbook on Ordinary Differential Equations: A Theoretical Approach
Contents
Preface
List of Figures
List of Tables
List of Abbreviations
1 Basic Concepts of Differential Equations
1.1 Introduction
1.1.1 Basic Concepts
1.2 Formation of Differential Equations
1.3 Classification of Solutions
1.4 Geometrical Interpretation of a Differential Equations of First Order and First Degree
1.5 Geometrical Classification of Solutions
1.5.1 Geometrical Interpretation of a Differential Equation of Second and Higher Order
1.6 Classification of Solutions
2 Uniform Convergence
2.1 Introduction
2.2 The Convergence of Sequences
2.3 The Weierstrass M Test
2.4 The Function of Two Variables: Lipschitz Condition
2.5 Lipschitz Condition
3 Existence and Uniqueness Theory
3.1 Introduction
3.2 Integral Equations Equivalent to IVPs
3.3 The Fundamental Existence and Uniqueness Theorem
3.4 Existence and Uniqueness Theorem
3.5 Picard’s Iteration Method
4 Nonlocal Existence Theorem
4.1 Introduction
4.2 Global Variant of the Existence and Uniqueness Theorem
4.3 Gronwall’s Integral Inequality
4.4 Continuity of Solutions
4.5 Dependence of Solution on Initial Conditions
4.6 Existence Theorem
4.6.1 Ascoli’s Lemma
4.7 Extremal Solutions
4.8 Lower and Upper Bound Solution
5 System of First-Order Differential Equations
5.1 Introduction
5.2 Differential Operators and an Operator Method
5.3 Linear Systems of Differential Equations
5.4 Differential Operator Method
5.5 An Operator Method for Linear Systems with Constant Coefficients
5.6 Homogenous Linear System with Constant Coefficients
5.7 Solution of Systems with Matrix Exponential
6 Non-Homogenous Linear Systems
6.1 Non-Homogenous Linear Systems
6.2 Solution of Non-Homogenous Differential Equations
6.3 Periodic Solutions of Linear System
6.4 Existence and Uniqueness Theorems for Linear Systems
6.5 Linear System in Vector Variables
6.6 Existence Theorems for Equations of Order n
7 Boundary Value Problems
7.1 Introduction
7.2 Sturm–Liouville Problems
7.3 Characteristic Value and Characteristic Function
7.4 Existence of Eigenvalues
7.5 Orthogonality of Eigenfunctions
8 Green’s Function and Sturm Theory
8.1 Introduction
8.2 Green’s Function
8.3 Green’s Function for Second-Order Equations
8.4 Construction of the Green’s Function
8.5 Construction of the Green’s Function for Second-Order Equations
8.5.1 Observation
9 Sturm Theory
9.1 Introduction
9.2 Self-Adjoint Equations of the Second Order
9.3 Some Basic Results of Sturm Theory
9.4 The Separation and Comparison Theorem
10 The Nonlinear Theory
10.1 Introduction
10.2 Elementary Critical Points for a System of Linear Equations
10.3 Classification of Critical Points
10.4 Critical Points and Stability for Linear Systems
10.5 Stability by Lyapunov’s Method
10.6 Simple Critical Points of Nonlinear Systems
11 Linearization
11.1 Introduction
11.2 Isolated Critical Points
11.3 Stability of Isolated Critical Points
11.4 The Trouble with Centers
11.5 Conservative Equations
12 Analytical and Numerical Methods for Differential Equations
12.1 Adomian Decomposition Method
12.2 Convergence Analysis of the Adomian Decomposition Method
12.3 Modified Adomian Decomposition Method
12.4 Application of Modified Adomian Decomposition Method
12.5 Numerical Examples
12.6 Homotopy Analysis Method
12.7 Basic Idea of Homotopy Analysis Method
12.8 Basic Idea of Homotopy Perturbation Method
12.9 Differential Transform Method
12.9.1 Differential Transform Method for Ordinary Differential Equations
12.9.2 Differential Transform Method for Partial Differential Equations
12.10 Convergence Analysis of Differential Transform Method
12.11 Operator in Differential Transform Method
12.12 Applications of the Differential Transform Method Ordinary Differential Equation
12.13 Solution of Ordinary Differential Equations
12.13.1 Euler’s Method
12.13.2 Runge–Kutta Method of Fourth Order
Index
About the Author
Back Cover
