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.
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
