Partial Differential Equations: Sources and Solutions

✍ Scribed by Arthur David Snider

• Publisher: Prentice Hall
• Year: 1999
• Tongue: English
• Leaves: 673
• Edition: 1
• Category: Library

✦ Synopsis

The author strikes a balance between rigor and ease of comprehension, while bravely confronting some mathematically untidy applications such as heat flow in a cylindrical wedge where the flat sides (rather than the cylindrical side) are heated externally. Topics include Fourier methods, differential equations of physics and engineering, methodology for the Sturm-Liouville sub-problems, Green's functions for non-homogeneous partial differential equations, and perturbation methods and their use in applied mathematics. Annotation c. by Book News, Inc., Portland, Or.

✦ Table of Contents

Contents
Preface
Chapter 1. Basics of Differential Equations
1.1 Review: Theory of Linear Ordinary Differential Equations
1.2 Change of Variable
1.3 Delta Functions
1.4 Green's Functions
1.5 Generalized Functions and Distributions
Chapter 2. Series Solutions for Ordinary Differential Equations
2.1 The Gamma Function
TABLE 2.1 Γ(x)
2.2 Taylor Series and Polynomials
TABLE 2.2: e^x= 1 + x + x^2 /2 + x^3 /6 + O(x^4)
2.3 Series Solutions at Ordinary Points
2.4 Frobenius's Method
2.5 Bessel Functions
TABLE 2.3 Bessel Function Properties
2.6 Famous Differential Equations
2.7 Numerical Solutions
Chapter 3. Fourier Methods
3.1 Oscillations
3.2 Transfer Functions
3.3 Fourier Series
3.4 Convergence of Fourier Series
3.5 Sine and Cosine Series
3.6 The Fourier Transform
3.7 Fourier Analysis of Generalized Functions
TABLE 3.1 FOURIER TRANSFORMS
3.8 The Laplace Transform
TABLE 3.2 LAPLACE TRANSFORMS
3.9 Frequency Domain and s-Plane Analysis
3.10 Numerical Fourier Analysis
Chapter 4. The Differential Equations of Physics and Engineering
4.1 The Calculus of Variations
4.2 Classical Mechanics
4.3 The Wave Equation in One Dimension
4.4 The Wave Equation in Higher Dimensions
4.5 The Heat Equation
4.6 Laplace's Equation
4.7 Schrödinger's Equation
4.8 Classification of Second Order Equations
Chapter 5. The Separation of Variables Technique
5.1 Overview of Separation of Variables
5.2 Separation of Variables: Basic Models
5.3 Separation of the Classical Equations
Chapter 6. Eigenfunction Expansions
6.1 Introduction
6.2 The Sturm-Liouville Theory
6.3 Regular Sturm-Liouville Expansions
6.4 Singular Sturm-Liouville Expansions
TABLE 6.1 Legendre Polynomials
6.5 Mathematical Issues
6.6 An Example for Reference
6.7 Derivation of Expansions for the Constant Coefficient Equation
6.8 Derivation of Expansions for the Equidimensional Equation
6.9 Derivations of Expansions for Bessel's Equation
6.10 The Associated Legendre Equation
TABLE 6.2 Associated Legendre Functions
TABLE 6.3 Eigenfunction Expansions
PART 1. Regular Sturm-Liouville Problems
PART 2. Singular Sturm-Liouville Problems
Chapter 7. Applications of Eigenfunctions to Partial Differential Equations
7.1 Eigenfunction Expansions in Higher Dimensions
TABLE 7.1 The Spherical Harmonics
7.2 Time-Dependent Problems: The Heat Equation
7.3 Initial Value Problems for The Wave Equation
7.4 The Schrödinger Equation
TABLE 7.2 Hermite Polynomials
TABLE 7.3 Laguerre Polynomials
Associated Laguerre Polynomials
7.5 Shortcuts
TABLE 7.4 Separated Solutions for Helmholtz's Equation ∇^2ψ = Λψ
Chapter 8. Green's Functions and Transform Methods
8.1 Expansions for Green's Functions
8.2 Transform Methods
8.3 Closed-Form Green's Functions
8.4 Discontinuities
8.5 Radiation Problems
Chapter 9. Perturbations, Small Waves, and Dispersion
9.1 Perturbation Methods for Algebraic Equations
9.2 Perturbation Methods for Differential Equations
9.3 Dispersion Laws and Wave Velocities
Appendix A. Some Numerical Techniques
A.1 Order of Magnitude
A.2 The Accuracy of Taylor Polynomials
A.3 Numerical Differentiation
A.4 Difference Quotients in Two Dimensions
A.5 Solution of Algebraic Equations
A.6 Numerical Integration
Appendix B. Evaluation of Bromwich Integrals
B.1 The Radiating Sphere
B.2 The Rectangular Waveguide
References
1-14
15-32
33-50
51-63
Index