Text presents the general properties of partial differential equations such as characteristics, domains of independence and maximum principles. Solutions.

Table of Contents for A First Course in Partial Differential Equations: with Complex Variables and Transform Methods 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

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