Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions

✍ Scribed by Victor Henner, Tatyana Belozerova, Kyle Forinash

• Publisher: A K Peters/CRC Press
• Year: 2009
• Tongue: English
• Leaves: 847
• Edition: 1
• Category: Library

✦ Synopsis

This book is a text on partial differential equations (PDEs) of mathematical physics and boundary value problems, trigonometric Fourier series, and special functions. This is the core content of many courses in the fields of engineering, physics, mathematics, and applied mathematics. The accompanying software provides a laboratory environment that allows the user to generate and model different physical situations and learn by experimentation. From this standpoint, the book along with the software can also be used as a reference book on PDEs, Fourier series and special functions for students and professionals alike.

✦ Table of Contents

Contents
Introduction
1. Fourier Series
1.1 Periodic Processes and Periodic Functions
1.2 Fourier Formulas
1.3 Orthogonal Systems of Functions
1.4 Convergence of Fourier Series
1.5 Fourier Series for Nonperiodic Functions
1.6 Fourier Expansions on Intervals of Arbitrary Length
1.7 Fourier Series in Cosine or Sine Functions
1.8 The Complex Form of the Fourier Series
1.9 Complex Generalized Fourier Series
1.10 Fourier Series for Functions of Several Variables
1.11 Uniform Convergence of Fourier Series
1.12 The Gibbs Phenomenon
1.13 Completeness of a System of Trigonometric Functions
1.14 General Systems of Functions: Parseval’s Equality and Completeness
1.15 Approximation of Functions in the Mean
1.16 Fourier Series of Functions Given at Discrete Points
1.17 Solution of Differential Equations by Using Fourier Series
1.18 Fourier Transforms
1.19 The Fourier Integral
Problems
2. Sturm-Liouville Theory
2.1 The Sturm-Liouville Problem
2.2 Mixed Boundary Conditions
2.3 Examples of Sturm-Liouville Problems
Problems
3. One-Dimensional Hyperbolic Equations
3.1 Derivation of the Basic Equations
3.2 Boundary and Initial Conditions
3.3 Other Boundary Value Problems: Longitudinal Vibrations of a Thin Rod
3.3.1 Derivation of the Basic Equations
3.3.2 Boundary Conditions for a Rod
3.4 Torsional Oscillations of an Elastic Cylinder
3.4.1 Derivation of the Basic Equation
3.4.2 Initial Conditions and Examples of Boundary Conditions for a Twisted Rod
3.5 Acoustic Waves
3.5.1 Derivation of the Basic Equation
3.5.2 Boundary Conditions for a Tube Filled with Gas
3.6 Waves in a Shallow Channel
3.6.1 Derivation of the Equations for Displacements ζ(x, t) and η(x, t)
3.6.2 Examples of Initial and Boundary Conditions
3.7 Electrical Oscillations in a Circuit
3.7.1 Derivation of the Basic Equations
3.7.2 Examples of Initial Conditions for Current, i(x, t)
3.7.3 Examples of Initial Conditions for Voltage, V (x, t)
3.7.4 Boundary Conditions for i(x, t) and V (x, t)
3.8 Traveling Waves: D’Alembert Method
3.8.1 The Infinite String
3.8.2 The Cauchy Problem
3.8.3 Nonhomogeneous Equations
3.8.4 Modifications of the General Form of the Wave Equation
3.9 Semi-infinite String Oscillations and the Use of Symmetry Properties
3.9.1 Boundary Value Problems with Homogeneous Boundary Conditions
3.9.2 Boundary Value Problems with Nonhomogeneous Boundary Conditions
3.10 Finite Intervals: The FourierMethod for One-Dimensional Wave Equations
3.10.1 The Fourier Method for Homogeneous Equations
3.10.2 The Fourier Method for Nonhomogeneous Equations
3.10.3 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
3.11 Generalized Fourier Solutions
3.12 Energy of the String
3.12.1 Kinetic and Potential Energies
3.12.2 Energy in the Harmonics
Problems
4. Two-Dimensional Hyperbolic Equations
4.1 Derivation of the Equations of Motion
4.1.1 Equations of Motion
4.1.2 Boundary and Initial Conditions
4.2 Oscillations of a Rectangular Membrane
4.2.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
4.2.2 The Fourier Method for Nonhomogeneous Equations
4.2.3 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
4.3 The FourierMethod Applied to Small Transverse Oscillations of a Circular Membrane
4.3.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
4.3.2 Radial Oscillations of a Membrane
4.3.3 The Fourier Method for Nonhomogeneous Equations
4.3.4 Radial Oscillations of a Circular Membrane
4.3.5 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
Problems
5. One-Dimensional Parabolic Equations
5.1 Physical Problems Described by Parabolic Equations: Boundary Value Problems
5.1.1 Heat Conduction
5.1.2 The Diffusion Equation
5.1.3 The One-Dimensional Parabolic Equation
5.1.4 Initial and Boundary Conditions
5.2 The Principle of the Maximum, Correctness, and the Generalized Solution
5.2.1 The Principle of the Maximum
5.3 The Fourier Method of Separation of Variables for the Heat Conduction Equation
5.3.1 A Simplification of the General Equation
5.3.2 The Fourier Method for Homogeneous Equations
5.3.3 The Fourier Method for Nonhomogeneous Equations
5.3.4 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
5.3.5 Boundary Problems without Initial Conditions
5.4 Heat Conduction in an Infinite Bar
5.5 Heat Equation for a Semi-infinite Bar
5.5.1 Boundary Value Problems on a Semi-infinite Interval with Homogeneous Boundary Conditions
5.5.2 Boundary Value Problems on a Semi-infinite Line with Nonhomogeneous Boundary Conditions
Problems
Propagation of Heat in an Infinite or Semi-infinite Rod
6. Parabolic Equations for Higher-Dimensional Problems
6.1 Heat Conduction in More than One Dimension
6.1.1 Heat Conduction in an Infinite Medium
6.1.2 Heat Conduction in a Semi-infinite Medium
6.2 Heat Conduction within a Finite Rectangular Domain
6.2.1 The Fourier Method for the Homogeneous Heat Conduction Equation (Free Heat Exchange within a Rectangular Plate)
6.2.2 The Fourier Method for the Nonhomogeneous Heat Conduction Equation (Rectangular Plate with Internal Sources)
6.2.3 The Fourier Method for the Nonhomogeneous Heat Conduction Equation with Nonhomogeneous Boundary Conditions
6.3 Heat Conduction within a Circular Domain
6.3.1 The Fourier Method for the Homogeneous Heat Conduction Equation (Free Heat Exchange within a Circular Plate)
6.3.2 The Fourier Method for the Nonhomogeneous Heat Conduction Equation (Circular Plates with Internal Sources)
6.3.3 The Fourier Method for the Nonhomogeneous Heat Conduction Equation with Nonhomogeneous Boundary Conditions
Problems
7. Elliptic Equations
7.1 Elliptic Partial Differential Equations and Related Physical Problems
7.1.1 Stationary Heat Conduction and Diffusion
7.1.2 Potential Fields and the Electrostatic Potential
7.1.3 Inviscid Flow of an Incompressible Fluid
7.1.4 Boundary Conditions: General Considerations
7.1.5 Principle of the Maximum and the Question of Well-Posed Boundary Value Problems
7.2 The Dirichlet Boundary Value Problem for Laplace’s Equation in a Rectangular Domain
7.2.1 A Method to Improve the Series Convergence when Boundary Conditions Match Each Other
7.2.2 Other Types of Boundary Conditions
7.2.3 Example of the Temperature Distribution in a Rectangular Domain
7.2.4 The Poisson Equation in a Rectangular Domain
7.3 Laplace’s and Poisson’s Equations for Two-Dimensional Domains with Circular Symmetry
7.3.1 The Fourier Method for Laplace’s Equation in Polar Coordinates
7.3.2 Laplace’s Equation for Interior Boundary Value Problems for a Circle
7.3.3 Laplace’s Equation for Exterior Boundary Value Problems for a Circle
7.3.4 Laplace’s Equation for Boundary Value Problems for an Annulus
7.3.5 Laplace’s Equation for Boundary Value Problems for a Circular Region
7.3.6 Example of Fluid Motion in a Cylinder as a Neumann Problem in a Plane
7.3.7 Poisson’s Equation: General Notes and a Simple Case
7.3.8 Poisson’s Equation: The General Case
7.3.9 Poisson’s Integral
7.4 Laplace’s Equation in Cylindrical Coordinates
7.4.1 Homogeneous Boundary Conditions at the Lateral Surface
7.4.2 Example of Homogeneous Boundary Conditions at the Lateral Surface
7.4.3 Homogeneous Boundary Conditions at the Bases
7.4.4 Example of Homogeneous Boundary Conditions at the Bases
7.4.5 Application of Bessel Functions for the Solution of Laplace’s and Poisson’s Equations in a Circle
7.4.6 Poisson’s Equation in Polar Coordinates with Homogeneous Boundary Conditions
Problems
The Interior Dirichlet Problem for a Circle
The Interior Neumann Problem for a Circle
Various Interior Problems for a Circle
Various Exterior Problems for a Circle
Various Problems for a Pie-Shaped Sector
Laplace’s Equation on a Ring
Two-Dimensional Poisson’s Problems
Truncation of a Series with a Given Accuracy
Laplace’s and Poisson’s Equations for a Cylinder
8. Bessel Functions
8.1 Boundary Value Problems Leading to Bessel Functions
8.2 Bessel Functions of the First Kind
8.3 Properties of Bessel Functions of the First Kind: Jn(x)
8.4 Bessel Functions of the Second Kind
8.5 Bessel Functions of the Third Kind
8.6 Modified Bessel Functions
8.7 The Effect of Boundaries on Bessel Functions
8.8 Orthogonality and Normalization of Bessel Functions
8.9 The Fourier-Bessel Series
8.10 Further Examples of Fourier-Bessel Series Expansions
8.11 Spherical Bessel Functions
8.12 The Gamma Function
Problems
9. Legendre Functions
9.1 Boundary Value Problems Leading to Legendre Polynomials
9.2 Generating Function for Legendre Polynomials
9.3 Recurrence Relations
9.4 Orthogonality of Legendre Polynomials
9.5 The Multipole Expansion in Electrostatics
9.6 Associated Legendre Functions P^m_n (x)
9.7 Orthogonality and the Normof Associated Legendre Functions
9.8 Fourier-Legendre Series in Legendre Polynomials
9.9 Fourier-Legendre Series in Associated Legendre Functions
9.10 Laplace’s Equation in Spherical Coordinates and Spherical Functions
9.10.1 Schrödinger Equation for a Central Potential
9.10.2 Oscillations of a Sphere: Sound Waves in a Spherical Cavity
9.10.3 Cooling of a Solid Sphere
Problems
A. Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem
B. Auxiliary Functions for Different Types of Boundary Conditions
C. The Sturm-Liouville Problem and the Laplace Equation
D. Vector Calculus
E. How to Use the Software Associated with this Book
E.1 Program Overview
E.2 Examples Using the Program TrigSeries
E.3 Examples Using the Program Waves
E.4 Examples Using the Program Heat
E.5 Examples Using the Program Laplace
E.6 Examples Using the Program FourierSeries
Bibliography