Solution Techniques for Elementary Partial Differential Equations

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Foreword
Preface to the Fourth Edition
Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
CHAPTER 1: ORDINARY DIFFERENTIAL EQUATIONS: BRIEF REVIEW
	1.1. First-Order Equations
	1.2. Homogeneous Linear Equations with Constant Coefficients
	1.3. Nonhomogeneous Linear Equations with Constant Coefficients
	1.4. Cauchy–Euler Equations
	1.5. Functions and Operators
CHAPTER 2: FOURIER SERIES
	2.1. The Full Fourier Series
	2.2. Fourier Sine and Cosine Series
		2.2.1. Fourier Sine Series
		2.2.2. Fourier Cosine Series
	2.3. Convergence and Differentiation
	2.4. Series Expansion of More General Functions
CHAPTER 3: STURM–LIOUVILLE PROBLEMS
	3.1. Regular Sturm–Liouville Problems
		3.1.1. Eigenvalues and Eigenfunctions of Some Basic Problems
		3.1.2. Generalized Fourier Series
	3.2. Other Problems
	3.3. Bessel Functions
	3.4. Legendre Polynomials
	3.5. Spherical Harmonics
CHAPTER 4: SOME FUNDAMENTAL EQUATIONS OF MATHEMATICAL PHYSICS
	4.1. The Heat Equation
	4.2. The Laplace Equation
	4.3. The Wave Equation
	4.4. Other Equations
CHAPTER 5: THE METHOD OF SEPARATION OF VARIABLES
	5.1. The Heat Equation
		5.1.1. Rod with Zero Temperature at the Endpoints
		5.1.2. Rod with Insulated Endpoints
		5.1.3. Rod with Mixed Boundary Conditions
		5.1.4. Rod with an Endpoint in a Zero-Temperature Medium
		5.1.5. Thin Uniform Circular Ring
	5.2. The Wave Equation
		5.2.1. Vibrating String with Fixed Endpoints
		5.2.2. Vibrating String with Free Endpoints
		5.2.3. Vibrating String with Other Types of Boundary Conditions
	5.3. The Laplace Equation
		5.3.1. Laplace Equation in a Rectangle
		5.3.2. Laplace Equation in a Circular Disc
	5.4. Other Equations
	5.5. Equations with More than Two Variables
		5.5.1. Vibrating Rectangular Membrane
		5.5.2. Vibrating Circular Membrane
		5.5.3. Equilibrium Temperature in a Solid Sphere
CHAPTER 6: LINEAR NONHOMOGENEOUS PROBLEMS
	6.1. Equilibrium Solutions
	6.2. Nonhomogeneous Problems
		6.2.1. Time-Independent Sources and Boundary Conditions
		6.2.2. The General Case
CHAPTER 7: THE METHOD OF EIGENFUNCTION EXPANSION
	7.1. The Nonhomogeneous Heat Equation
		7.1.1. Rod with Zero Temperature at the Endpoints
		7.1.2. Rod with Insulated Endpoints
		7.1.3. Rod with Mixed Boundary Conditions
	7.2. The Nonhomogeneous Wave Equation
		7.2.1. Vibrating String with Fixed Endpoints
		7.2.2. Vibrating String with Free Endpoints
		7.2.3. Vibrating String with Mixed Boundary Conditions
	7.3. The Nonhomogeneous Laplace Equation
		7.3.1. Equilibrium Temperature in a Rectangle
		7.3.2. Equilibrium Temperature in a Circular Disc
	7.4. Other Nonhomogeneous Equations
CHAPTER 8: THE FOURIER TRANSFORMATIONS
	8.1. The Full Fourier Transformation
		8.1.1. Cauchy Problem for an Infinite Rod
		8.1.2. Vibrations of an Infinite String
		8.1.3. Equilibrium Temperature in an Infinite Strip
	8.2. The Fourier Sine and Cosine Transformations
		8.2.1. Heat Conduction in a Semi-Infinite Rod
		8.2.2. Vibrations of a Semi-Infinite String
		8.2.3. Equilibrium Temperature in a Semi-Infinite Strip
	8.3. Other Applications
CHAPTER 9: THE LAPLACE TRANSFORMATION
	9.1. Definition and Properties
	9.2. Applications
		9.2.1. The Signal Problem for the Wave Equation
		9.2.2. Heat Conduction in a Semi-Infinite Rod
		9.2.3. Finite Rod with Temperature Prescribed on the Boundary
		9.2.4. Diffusion–Convection Problems
		9.2.5. Dissipative Waves
CHAPTER 10: THE METHOD OF GREEN'S FUNCTIONS
	10.1. The Heat Equation
		10.1.1. The Time-Independent Problem
		10.1.2. The Time-Dependent Problem
	10.2. The Laplace Equation
	10.3. The Wave Equation
CHAPTER 11: GENERAL SECOND-ORDER LINEAR EQUATIONS
	11.1. The Canonical Form
		11.1.1. Classification
		11.1.2. Reduction to Canonical Form
	11.2. Hyperbolic Equations
	11.3. Parabolic Equations
	11.4. Elliptic Equations
	11.5. Other Problems
CHAPTER 12: THE METHOD OF CHARACTERISTICS
	12.1. First-Order Linear Equations
	12.2. First-Order Quasilinear Equations
	12.3. The One-Dimensional Wave Equation
		12.3.1. The d’Alembert Solution
		12.3.2. The Semi-Infinite Vibrating String
		12.3.3. The Finite String
	12.4. Other Hyperbolic Equations
		12.4.1. One-Dimensional Waves
		12.4.2. Moving Boundary Problems
		12.4.3. Spherical Waves
CHAPTER 13: PERTURBATION AND ASYMPTOTIC METHODS
	13.1. Asymptotic Series
	13.2. Regular Perturbation Problems
		13.2.1. Formal Solutions
		13.2.2. Secular Terms
	13.3. Singular Perturbation Problems
CHAPTER 14: COMPLEX VARIABLE METHODS
	14.1. Elliptic Equations
	14.2. Systems of Equations
Appendix
	A.1. Useful Integrals
	A.2. Eigenvalue–Eigenfunction Pairs
	A.3. Table of Fourier Transforms
	A.4. Table of Fourier Sine Transforms
	A.5. Table of Fourier Cosine Transforms
	A.6. Table of Laplace Transforms
	A.7. Second-Order Linear Equations
Further Reading
Index