Ordinary Differential Equations and Special Functions

1. Auflage 2025
ISBN: 978-1-394-38503-4
Verlag: Wiley

Buch, Englisch, 640 Seiten, Gewicht: 975 g

This book is an essential guide for anyone in engineering or mathematical physics looking to master the fundamental concepts of differential equations and special functions, which are crucial for solving real-world problems.

In today’s evolving mathematics landscape, differential equations and special functions have shown great promise for applications in engineering. Problems in mathematical physics help determine solutions for differential equations under certain parameters, which can be turned into new special functions, such as Bessel’s functions, to measure electricity, hydrodynamics, and vibration. Ordinary Differential Equations and Special Functions serves as a fundamental guide to these concepts, covering everything from elementary-level special functions to differential equations with a series of solutions.

Autoren/Hrsg.

De, Dipankar

Fachgebiete

• Technische Wissenschaften Energietechnik | Elektrotechnik Elektrotechnik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Preface xiii
Introduction xv
About the Book xvii

Part I: Ordinary Differential Equations 1

1 Preliminaries I 3
1.1 Introduction 3
1.2 Formation of a Differential Equation 4
1.3 Family of Curves Represented by Ordinary Differential Equations 8
1.4 Equation of the First Order and First Degree 11
1.5 Equations of the First Order and Higher Degree 24
1.6 Linear Differential Equation 30
1.7 Other Methods of Finding P.I. 40
1.8 Differential Equation of Other Types 44
1.9 Orthogonal Trajectories 51
1.10 Examples 52
1.11 Exercise 63

2 Existence Theorems 67
2.1 Introduction 67
2.2 Initial Value Problems and Boundary Value Problems 69
2.3 Picard's Method of Successive Approximation 70
2.4 Lipschitz Condition 78
2.5 Picard's Theorem: Existence and Uniqueness Theorem 80
2.6 Singular Solutions 90
2.7 Clairaut Equation 92
2.8 Examples 95
2.9 Exercise 99

3 System of Linear Differential Equations-I 101
3.1 Introduction 101
3.2 Matrix Form of a Linear System 102
3.3 Reduction of an nth-Order Equation 104
3.4 Matrix Preliminaries 107
3.5 Fundamental Set of Solutions 109
3.6 Solution of Non-Homogeneous Linear Systems 126
3.7 Linear System with Constant Coefficients 130
3.8 Exercise 138

4 Systems of Linear Differential Equations-II 141
4.1 Introduction 141
4.2 Linearly Dependent and Independent Functions 144
4.3 The Second-Order Homogeneous Equation 151
4.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters 157
4.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients 163
4.6 Examples 174
4.7 Exercise 177

5 Adjoint Equation 181
5.1 Introduction 181
5.2 Adjoint Equation 181
5.3 Green's Formula 194
5.4 Examples 196
5.5 Exercise 201

6 Boundary Value Problem 203
6.1 Introduction 203
6.2 Green's Function 207
6.3 Examples 210
6.4 Exercise 215

7 Strum Liouville Problem 217
7.1 Introduction 217
7.2 Strum–Liouville Equation 217

Über Autor(innen)

Dipankar De, PhD is an contractual professor and guest lecturer with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.