Partial Differential Equations

Partial Differential Equations: Theory and Technique provides formal definitions, notational conventions, and a systematic discussion of partial differential equations.

The text emphasizes the acquisition of practical technique in the use of partial differential equations. The book contains discussions on classical second-order equations of diffusion, wave motion, first-order linear and quasi-linear equations, and potential theory. Certain chapters elaborate Green's functions, eigenvalue problems, practical approximation techniques, perturbations (regular and singular), difference equations, and numerical methods.

Students of mathematics will find the book very useful.

Preface

Introduction

1.1 Definitions and Examples

Chapter 1 The Diffusion Equation

1.1 Derivation

1.2 Problems

1.3 Simple Solutions

1.4 Problems

1.5 Series Solutions

1.6 Problems

1.7 Nonhomogeneous End Conditions

1.8 Problems

1.9 The Maximum Principle

1.10 Problems

Chapter 2 Laplace Transform Methods

2.1 Introductory Example

2.2 Problems

2.3 A Finite Interval Problem

2.4 Problems

2.5 Delta Function

2.6 Problems

2.7 Supplementary Problems

Chapter 3 The Wave Equation

3.1 Derivation

3.2 Problems

3.3 An Infinite-Interval Problem

3.4 Problems

3.5 Series Solutions

3.6 Problems

3.7 A Problem with Radial Symmetry

3.8 Problems

3.9 Transforms

3.10 Problems

3.11 Uniqueness

3.12 Supplementary Problems

Chapter 4 The Potential Equation

4.1 Laplace's and Poisson's Equations

4.2 Problems

4.3 Simple Properties of Harmonic Functions

4.4 Some Special Solutions—Series

4.5 Problems

4.6 Discontinuous Boundary Data

4.7 Complex Variables and Conformai Mapping

4.8 Problems

Chapter 5 Classification of Second-Order Equations

5.1 Cauchy Data on y-Axis

5.2 Cauchy Data on Arbitrary Curve

5.3 Problems

5.4 Case I: B2 - AC > 0

5.5 Case II: B2 - AC = 0

5.6 Case III: J52 - AC < 0

5.7 Problems

5.8 Discontinuities; Signal Propagation

5.9 Problems

5.10 Some Remarks

Chapter 6 First-Order Equations

6.1 Linear Equation Examples

6.2 Problems

6.3 Quasi-Linear Case

6.4 Problems

6.5 Further Properties of Characteristics

6.6 Problems

6.7 More Variables

Chapter 7 Extensions

7.1 More Variables

7.2 Problems

7.3 Series and Transforms

7.4 Problems

7.5 Legendre Functions

7.6 Problems

7.7 Spherical Harmonics

7.8 Problems

Chapter 8 Perturbations

8.1 A Nonlinear Problem

8.2 Problems

8.3 Two Examples from Fluid Mechanics

8.4 Boundary Perturbations

8.5 Problems

Chapter 9 Green's Functions

9.1 Some Consequences of the Divergence Theorem

9.2 The Laplacian Operator

9.3 Problems

9.4 Potentials of Volume and Surface Distributions

9.5 Problems

9.6 Modified Laplacian

9.7 Problems

9.8 Wave Equation

9.9 Problems

Chapter 10 Variational Methods

10.1 A Minimization Problem

10.2 Problems

10.3 Natural Boundary Conditions

10.4 Subsidiary Conditions

10.5 Problems

10.6 Approximate Methods

10.7 Problems

10.8 Finite-Element Method

10.9 Supplementary Problems

Chapter 11 Eigenvalue Problems

11.1 A Prototype Problem

11.2 Some Eigenvalue Properties

11.3 Problems

11.4 Perturbations

11.5 Approximations

11.6 Problems

Chapter 12 More on First-Order Equations

12.1 Envelopes

12.2 Characteristic Strips

12.3 Complete Integral

12.4 Problems

12.5 Legendre Transformation

12.6 Problems

12.7 Propagation of a Disturbance

12.8 Complete Integral and Eikonal Function

12.9 Hamilton-Jacobi Equation

12.10 Problems

Chapter 13 More on Characteristics

13.1 Discontinuities—A Preliminary Example

13.2 Weak Solutions

13.3 Burgers' Equation

13.4 Problems

13.5 A Compressible Flow Problem

13.6 A Numerical Approach

13.7 Problems

13.8 More Dependent Variables

13.9 More Independent Variables

13.10 Problems

Chapter 14 Finite-Difference Equations and Numerical Methods

14.1 Accuracy and Stability; A Diffusion Equation Example

14.2 Error Analysis

14.3 Problems

14.4 More Dimensions, or Other Complications

14.5 Series Expansions

14.6 Problems

14.7 Wave Equation

14.8 A Nonlinear Equation

14.9 Problems

14.10 Boundary Value Problems

14.11 Problems

14.12 Series; Fast Fourier Transform

14.13 Problems

Chapter 15 Singular Perturbation Methods

15.1 A Boundary Layer Problem

15.2 A More General Procedure

15.3 Problems

15.4 A Transition Situation

15.5 Problems

15.6 Asymptotic Analysis of Wave Motion

15.7 Boundary Layer near a Caustic

15.8 Problems

15.9 Multiple Scaling

15.10 Problems

References

Index