A First Course in Partial Differential Equations: with Complex Variables and Transform Methods

by H. F. Weinberger

Overview

This popular text was created for a one-year undergraduate course or beginning graduate course in partial differential equations, including the elementary theory of complex variables. It employs a framework in which the general properties of partial differential equations, such as characteristics, domains of independence, and maximum principles, can be clearly seen. The only prerequisite is a good course in calculus.
Incorporating many of the techniques of applied mathematics, the book also contains most of the concepts of rigorous analysis usually found in a course in advanced calculus. These techniques and concepts are presented in a setting where their need is clear and their application immediate. Chapters I through IV cover the one-dimensional wave equation, linear second-order partial differential equations in two variables, some properties of elliptic and parabolic equations and separation of variables, and Fourier series. Chapters V through VIII address nonhomogeneous problems, problems in higher dimensions and multiple Fourier series, Sturm-Liouville theory, and general Fourier expansions and analytic functions of a complex variable.
The last four chapters are devoted to the evaluation of integrals by complex variable methods, solutions based on the Fourier and Laplace transforms, and numerical approximation methods. Numerous exercises are included throughout the text, with solutions at the back.

Product Details

ISBN-13:	9780486686400
Publisher:	Dover Publications
Publication date:	09/11/1995
Series:	Dover Books on Mathematics
Edition description:	Dover ed
Pages:	480
Sales rank:	874,533
Product dimensions:	6.14(w) x 9.21(h) x (d)

Table of Contents

I. The one-dimensional wave equation
1. A physical problem and its mathematical models: the vibrating string
2. The one-dimensional wave equation
3. Discussion of the solution: characteristics
4. Reflection and the free boundary problem
5. The nonhomogeneous wave equation
II. Linear second-order partial differential equations in two variables
6. Linearity and superposition
7. Uniqueness for the vibrating string problem
8. Classification of second-order equations with constant coefficients
9. Classification of general second-order operators
III. Some properties of elliptic and parabolic equations
10. Laplace's equation
11. Green's theorem and uniqueness for the Laplace's equation
12. The maximum principle
13. The heat equation
IV. Separation of variables and Fourier series
14. The method of separation of variables
15. Orthogonality and least square approximation
16. Completeness and the Parseval equation
17. The Riemann-Lebesgue lemma
18. Convergence of the trigonometric Fourier series
19. "Uniform convergence, Schwarz's inequality, and completeness"
20. Sine and cosine series
21. Change of scale
22. The heat equation
23. Laplace's equation in a rectangle
24. Laplace's equation in a circle
25. An extension of the validity of these solutions
26. The damped wave equation
V. Nonhomogeneous problems
27. Initial value problems for ordinary differential equations
28. Boundary value problems and Green's function for ordinary differential equations
29. Nonhomogeneous problems and the finite Fourier transform
30. Green's function
VI. Problems in higher dimensions and multiple Fourier series
31. Multiple Fourier series
32. Laplace's equation in a cube
33. Laplace's equation in a cylinder
34. The three-dimensional wave equation in a cube
35. Poisson's equation in a cube
VII. Sturm-Liouville theory and general Fourier expansions
36. Eigenfunction expansions for regular second-order ordinary differential equations
37. Vibration of a variable string
38. Some properties of eigenvalues and eigenfunctions
39. Equations with singular endpoints
40. Some properties of Bessel functions
41. Vibration of a circular membrane
42. Forced vibration of a circular membrane: natural frequencies and resonance
43. The Legendre polynomials and associated Legendre functions
44. Laplace's equation in the sphere
45. Poisson's equation and Green's function for the sphere
VIII. Analytic functions of a complex variable
46. Complex numbers
47. Complex power series and harmonic functions
48. Analytic functions
49. Contour integrals and Cauchy's theorem
50. Composition of analytic functions
51. Taylor series of composite functions
52. Conformal mapping and Laplace's equation
53. The bilinear transformation
54. Laplace's equation on unbounded domains
55. Some special conformal mappings
56. The Cauchy integral representation and Liouville's theorem
IX. Evaluation of integrals by complex variable methods
57. Singularities of analytic functions
58. The calculus of residues
59. Laurent series
60. Infinite integrals
61. Infinite series of residues
62. Integrals along branch cuts
X. The Fourier transform
63. The Fourier transform
64. Jordan's lemma
65. Schwarz's inequality and the triangle inequality for infinite integrals
66. Fourier transforms of square integrable functions: the Parseval equation
67. Fourier inversion theorems
68. Sine and cosine transforms
69. Some operational formulas
70. The convolution product
71. Multiple Fourier transforms: the heat equation in three dimensions
72. The three-dimensional wave equation
73. The Fourier transform with complex argument
XI. The Laplace transform
74. The Laplace transform
75. Initial value problems for ordinary differential equations
76. Initial value problems for the one-dimensional heat equation
77. A diffraction problem
78. The Stokes rule and Duhamel's principle
XII. Approximation methods
79. "Exact" and approximate solutions"
80. The method of finite differences for initial-boundary value problems
81. The finite difference method for Laplace's equation
82. The method of successive approximations
83. The Raleigh-Ritz method
SOLUTIONS TO THE EXERCISES
INDEX