Description
Dover A First Course in Partial Differential Equations with Complex Variables and Transform Methods 1995 Edition by Hans F. Weinberger
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. Table of contents :- I. The one-dimensional wave equation1. A physical problem and its mathematical models: the vibrating string2. The one-dimensional wave equation3. Discussion of the solution: characteristics4. Reflection and the free boundary problem5. The nonhomogeneous wave equationII. Linear second-order partial differential equations in two variables6. Linearity and superposition7. Uniqueness for the vibrating string problem8. Classification of second-order equations with constant coefficients9. Classification of general second-order operatorsIII. Some properties of elliptic and parabolic equations10. Laplace's equation11. Green's theorem and uniqueness for the Laplace's equation12. The maximum principle13. The heat equationIV. Separation of variables and Fourier series14. The method of separation of variables15. Orthogonality and least square approximation 16. Completeness and the Parseval equation17. The Riemann-Lebesgue lemma18. Convergence of the trigonometric Fourier series19. "Uniform convergence, Schwarz's inequality, and completeness"20. Sine and cosine series21. Change of scale22. The heat equation23. Laplace's equation in a rectangle24. Laplace's equation in a circle25. An extension of the validity of these solutions26. The damped wave equationV. Nonhomogeneous problems27. Initial value problems for ordinary differential equations28. Boundary value problems and Green's function for ordinary differential equations29. Nonhomogeneous problems and the finite Fourier transform30. Green's functionVI. Problems in higher dimensions and multiple Fourier series31. Multiple Fourier series32. Laplace's equation in a cube33. Laplace's equation in a cylinder34. The three-dimensional wave equation in a cube35. Poisson's equation in a cubeVII. Sturm-Liouville theory and general Fourier expansions36. Eigenfunction expansions for regular second-order ordinary differential equations37. Vibration of a variable string38. Some properties of eigenvalues and eigenfunctions39. Equations with singular endpoints40. Some properties of Bessel functions41. Vibration of a circular membrane42. Forced vibration of a circular membrane: natural frequencies and resonance43. The Legendre polynomials and associated Legendre functions44. Laplace's equation in the sphere45. Poisson's equation and Green's function for the sphereVIII. Analytic functions of a complex variable46. Complex numbers47. Complex power series and harmonic functions48. Analytic functions49. Contour integrals and Cauchy's theorem50. Composition of analytic functions51. Taylor series of composite functions52. Conformal mapping and Laplace's equation53. The bilinear transformation54. Laplace's equation on unbounded domains55. Some special conformal mappings56. The Cauchy integral representation and Liouville's theoremIX. Evaluation of integrals by complex variable methods57. Singularities of analytic functions58. The calculus of residues59. Laurent series60. Infinite integrals61. Infinite series of residues62. Integrals along branch cutsX. The Fourier transform63. The Fourier transform64. Jordan's lemma65. Schwarz's inequality and the triangle inequality for infinite integrals66. Fourier transforms of square integrable functions: the Parseval equation67. Fourier inversion theorems68. Sine and cosine transforms69. Some operational formulas70. The convolution product71. Multiple Fourier transforms: the heat equation in three dimensions72. The three-dimensional wave equation 73. The Fourier transform with complex argumentXI. The Laplace transform74. The Laplace transform75. Initial value problems for ordinary differential equations76. Initial value problems for the one-dimensional heat equation77. A diffraction problem78. The Stokes rule and Duhamel's principleXII. Approximation methods79. "Exact" and approximate solutions"80. The method of finite differences for initial-boundary value problems81. The finite difference method for Laplace's equation82. The method of successive approximations83. The Raleigh-Ritz methodSOLUTIONS TO THE EXERCISESINDEX