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A First Course in Partial Differential Equations with Complex Variables and Tran (Dover Books on Mathematics)


A First Course in Partial Differential Equations with Complex Variables and Tran (Dover Books on Mathematics) Cover

ISBN13: 9780486686400
ISBN10: 048668640x
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Synopses & Reviews

Publisher Comments:

Suitable for advanced undergraduate and graduate students, this text presents the general properties of partial differential equations, including the elementary theory of complex variables. Topics include one-dimensional wave equation, properties of elliptic and parabolic equations, separation of variables and Fourier series, nonhomogeneous problems, and analytic functions of a complex variable. Solutions. 1965 edition.


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

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



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the book has got a title which makes a student beginning the want to go for it.
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Product Details

Weinberger, H. F.
Weinberger, Hans F.
Dover Publications
New York :
Differential Equations
Algebra - General
Differential Equations - Partial Differential Equations
Differential equations, partial
General Mathematics
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Dover Books on Mathematics
Publication Date:
9.25 x 6.25 in 1.44 lb

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Related Subjects

» Science and Mathematics » Mathematics » Calculus » General
» Science and Mathematics » Mathematics » Differential Equations
» Science and Mathematics » Mathematics » General
» Science and Mathematics » Mathematics » Topology

A First Course in Partial Differential Equations with Complex Variables and Tran (Dover Books on Mathematics) New Trade Paper
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Product details 480 pages Dover Publications - English 9780486686400 Reviews:
"Synopsis" by ,
Text presents the general properties of partial differential equations such as characteristics, domains of independence, and maximum principles. Solutions.
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