This book highlights a concise and readable introduction to typical treatments of partial differential equations in mathematical physics. Mathematical physics is regarded by many as a profound discipline. In conventional textbooks of mathematical physics, the known and the new pieces of knowledge often intertwine with each other. The book aims to ease readers' struggle by facilitating a smooth transition to new knowledge. To achieve so, the author designs knowledge maps before each chapter and provides comparative summaries in each chapter whenever appropriate. Through these unique ways, readers can clarify the underlying structures among different equations and extend one's vision to the big picture. The book also emphasizes applications of the knowledge by providing practical examples. The book is intended for all those interested in mathematical physics, enabling them to develop a solid command in using partial differential equations to solve physics and engineering problems in a not-so-painful learning experience.
Author(s): Yichao Zhu
Publisher: Springer
Year: 2021
Language: English
Pages: 255
Preface
Contents
Part I Second-Order Linear Partial Differential Equations
1 The Wave Equation
1.1 Equation for String Vibration
1.1.1 Derivation of the Equation for String Vibration
1.1.2 Initial and Boundary Conditions
1.1.3 Terminology
1.2 D'Alembert's Formula
1.2.1 D'Alembert's Formula
1.2.2 Characteristics
1.2.3 The Case of a Semi-infinitely Long String
1.2.4 Duhamel's Principle
1.3 Method of Separation of Variables
1.3.1 Making Boundary Conditions Homogeneous
1.3.2 Method of Separation of Variables—Its Procedure
1.3.3 Physical Implications
1.3.4 Inhomogeneous Governing Equations
1.4 Wave Equation in Higher Dimensions
1.4.1 Small and Transverse Vibration of a Membrane
1.4.2 Definite Problems
1.4.3 Solutions for Cauchy Problems
1.5 Solution Properties
1.5.1 The Energy of a Vibrating Membrane
1.5.2 Solution Uniqueness of Problems for the Wave Equation
2 The Heat Equation
2.1 Modelling Heat Conduction
2.1.1 Derivation of the Heat Equation
2.1.2 Initial and Boundary Conditions
2.1.3 Physical Analogies
2.2 Method of Integral Transform
2.2.1 Convolution and Fourier Transform
2.2.2 Solution for Cauchy Problems
2.2.3 Solution Properties
2.2.4 Inhomogeneous Governing Equations
2.3 A Revisit to the Method of Separation of Variables
2.3.1 An Example with the Heat Equation
2.3.2 Sturm–Liouville System
2.3.3 Inhomogeneous Governing Equations
2.4 Solution Properties
2.4.1 Maximum Principle
2.4.2 Solution Uniqueness
2.4.3 Stability
3 Poisson's Equation
3.1 Poisson's Equation and Harmonic Equation
3.1.1 Definitions
3.1.2 Motivation from Physics
3.1.3 Boundary Conditions
3.2 Variational Principle
3.3 Harmonic Functions in Polar System
3.3.1 Laplace's Equation in Polar System
3.3.2 Radial Solutions to Laplace's Equation
3.4 The Method of Green's Function
3.4.1 Green's Formulae Related to Laplacian Operator
3.4.2 Fundamental Solution
3.4.3 Derivation of Green's Function
3.4.4 Properties of Green's Function
3.4.5 Problems for Poisson's Equation
3.4.6 Final Remarks
3.5 Image Method for Electric Potentials
3.5.1 The Case in Three-Dimensional Half Space
3.5.2 The Case in a Spherical Domain
3.6 Solution Uniqueness
3.6.1 Mean-Value Formula
3.6.2 Maximum Principle
3.6.3 Strong Maximum Principle
3.6.4 Energy Method
4 Summary over Second-Order Linear Partial Differential Equation
4.1 Classification of Second-Order Linear Partial Differential Equations
4.1.1 Cases with Two Variables
4.1.2 Summary and Examples
4.1.3 Multivariable Situations
4.2 Topical Discussion
Part II Special Functions
5 Bessel Functions
5.1 Bessel Equation and Bessel Functions
5.1.1 Physical Motivation
5.1.2 Bessel Function of the First Kind
5.1.3 Bessel Function of the Second Kind
5.2 Properties of Bessel Functions
5.2.1 Recurrence Formulae
5.2.2 Zeros
5.2.3 Approximating Formula
5.2.4 Orthogonality
5.2.5 Analogies with Sinusoidal Functions
5.3 Solving PDE Problems with Bessel Functions
5.4 Generalisation
5.4.1 Hankel Functions
5.4.2 Modified Bessel Functions
6 Legendre Polynomial
6.1 Legendre Equation
6.2 Legendre Polynomial
6.2.1 Series Solution to the Legendre Equation
6.2.2 Legendre Polynomial
6.3 Properties of Legendre Polynomial
6.3.1 Rodrigues Formula
6.3.2 Key Properties at a Glance
6.3.3 Orthogonal Systems
6.3.4 Discussion on General Orthogonal Polynomials
6.4 Applications with Legendre Polynomials
6.4.1 Solving PDEs Defined in a Sphere
6.4.2 Legendre–Gauss Quadrature
6.5 The Associated Legendre Functions
7 Introduction of Hypergeometric Function
7.1 Commonalities of Bessel and Legendre Functions
7.2 Gauss Hypergeometric Function
7.2.1 Definition
7.2.2 Hypergeometric Differential Equation
7.2.3 Legendre Function and Legendre Polynomial—A Revisit
7.3 Confluent Hypergeometric Function
7.3.1 Kummer Differential Equation and Its Solutions
7.3.2 Special Cases
7.4 Final Remarks on Hypergeometric Functions
Appendix Bibliography
Index
