Solution Techniques for Elementary Partial Differential Equations

by Constanda; Christian

• ISBN13: 9781584882572
• ISBN10: 1584882573
• Edition: 1st
• Format: Paperback
• Copyright: 2002-02-26
• Publisher: Chapman & Hall/

Summary

Of the many available texts on partial differential equations (PDEs), most are too detailed and voluminous, making them daunting to many students. In sharp contrast, Solution Techniques for Elementary Partial Differential Equations is a no-frills treatment that explains completely but succinctly some of the most fundamental solution methods for PDEs. After a brief review of elementary ODE techniques and discussions on Fourier series and Sturm-Liouville problems, the author introduces the heat, Laplace, and wave equations as mathematical models of physical phenomena. He then presents a number of solution techniques and applies them to specific initial/boundary value problems for these models. Discussion of the general second order linear equation in two independent variables follows, and finally, the method of characteristics and perturbation methods are presented. Most students seem to like concise, easily digestible explanations and worked examples that let them see the techniques in action. This text offers them both. Ideally suited for independent study and classroom tested with great success, it offers a direct, streamlined route to competence in PDE solution techniques.

Table of Contents

Foreword	xi
Preface	xiii
Ordinary Differential Equations: Brief Revision	1	(10)
First-Order Equations	1	(2)
Homogeneous Linear Equations with Constant Coefficients	3	(2)
Nonhomogeneous Linear Equations with Constant Coefficients	5	(1)
Linear Operators	6	(5)
Exercises	8	(3)
Fourier Series	11	(18)
The Full Fourier Series	11	(6)
Fourier Sine Series	17	(4)
Fourier Cosine Series	21	(3)
Convergence and Differentiation	24	(5)
Exercises	25	(4)
Sturm-Liouville Problems	29	(20)
Regular Sturm-Liouville Problems	29	(11)
Other Sturm-Liouville Problems	40	(9)
Exercises	44	(5)
Three Fundamental Equations of Mathematical Physics	49	(20)
The Heat Equation	49	(8)
The Laplace Equation	57	(6)
The Wave Equation	63	(6)
The Method of Separation of Variables	69	(42)
The Heat Equation	69	(13)
The Wave Equation	82	(6)
The Laplace Equation	88	(8)
Equations with More than Two Variables	96	(15)
Exercises	104	(7)
Linear Nonhomogeneous Problems	111	(12)
Equilibrium Solutions	111	(5)
Nonhomogeneous Problems	116	(7)
Exercises	120	(3)
The Method of Eigenfunction Expansion	123	(18)
The Heat Equation	123	(6)
The Wave Equation	129	(3)
The Laplace Equation	132	(9)
Exercises	135	(6)
The Fourier Transformations	141	(16)
The Full Fourier Transformation	141	(6)
The Fourier Sine and Cosine Transformations	147	(10)
Exercises	154	(3)
The Laplace Transformation	157	(18)
Definition and Properties	157	(5)
Applications	162	(13)
Exercises	172	(3)
The Method of Green's Functions	175	(20)
The Heat Equation	175	(7)
The Laplace Equation	182	(3)
The Wave Equation	185	(10)
Exercises	191	(4)
General Second-Order Linear Partial Differential Equations with Two Independent Variables	195	(12)
The Canonical Form	195	(4)
Hyperbolic Equations	199	(3)
Parabolic Equations	202	(2)
Elliptic Equations	204	(3)
Exercises	205	(2)
The Method of Characteristics	207	(18)
First-Order Linear Equations	207	(7)
First-Order Quasilinear Equations	214	(1)
The One-Dimensional Wave Equation	215	(10)
Exercises	222	(3)
Perturbation and Asymptotic Methods	225	(18)
Asymptotic Series	225	(3)
Regular Perturbation Problems	228	(6)
Singular Perturbation Problems	234	(9)
Exercises	240	(3)
Appendix	243	(4)
Bibliography	247	(2)
Index	249