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PARTIAL DIFFERENTIAL EQUATIONS AN INTRODUCTION
  • BERNARD EPSTEIN 著
  • 出版社： INC.
  • ISBN：
  • 出版时间：1962
  • 标注页数：273页
  • 文件大小：12MB
  • 文件页数：282页
  • 主题词：

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图书目录

CHAPTER 1．Some Preliminary Topics1

1．Equicontinuous Families of Functions1

2．The Weierstrass Approximation Theorem4

3．The Fourier Integral9

4．The Laplace Transform13

5．Ordinary Differential Equations17

6．Lebesgue Integration25

7．Dini＇s Theorem27

CHAPTER 2．Partial Differential Equations of First Order28

1．Linear Equations in Two Independent Variables28

2．Quasi-linear Equations33

3．The General First-order Equation36

CHAPTER 3．The Cauchy Problem42

1．Classification of Equations with Linear Principal Parts42

2．Characteristics44

3．Canonical Forms46

4．The Cauchy Problem for Hyperbolic Equations48

5．The One-dimensional Wave Equation53

6．The Riamann Function55

7．Classification of Second-order Equations in Three or More Independent Variables58

8．The Wave Equation in Two and Three Dimensions60

9．The Legendre Transformation65

CHAPTER 4．The Fredholm Alternative in Banach Spaces69

1．Linear Spaces69

2．Normed Linear Spaces71

3．Banach Spaces74

4．Linear Functionals and Linear Operators76

5．The Fredholm Alternative82

CHAPTER 5．The Fredholm Alternative Hilbert Spaces90

1．Inner-product Spaces90

2．Hilbert Spaces95

3．Projections,Linear Functionals,Adjoint Operators99

4．Hermitian and Completely Continuous Operators104

5．The Fredholm Alternative111

6．Integral Equations118

7．Hermitian Kernels121

8．Illustrative Example127

CHAPTER 6．Elements of Potential Theory130

1．Introduction130

2．Laplace＇s Equation and Theory of Analytic Functions131

3．Fundamental Solutions133

4．The Mean-value Theorem135

5．The Maximum Principle136

6．Formulation of the Dirichlet Problem138

7．Solution of the Dirichlet Problem for the Disc139

8．The Converse of the Mean-value Theorem146

9．Convergence Theorems149

10．Strengthened Form of the Maximum Principle152

11．Single and Double Layers152

12．Poisson＇s Equation157

CHAPTER 7．The Dirichlet Problem167

1．Subharmonic Functions167

2．The Method of Balayage170

3．The Perron-Remak Method176

4．The Method of Integral Equations179

5．The Dirichlet Principle183

6．The Method of Finite Differences199

7．Conformal Mapping211

CHAPTER 8．The Heat Equation217

1．The Initial-value Problem for the Infinite Rod217

2．The Simplest Problem for the Semi-infinite Rod221

3．The Finite Rod256

CHAPTER 9．Green＇s Functions and Separation of Variables232

1．The Vibrating String232

2．The Green＇s Function of the Operator d2/dx2235

3．The Green＇s Function of a Second-order Differential Operator237

4．Eigenfunction Expansions239

5．A Generalized Wave Equation241

6．Extension of the Definition of Green＇s Functions243

SOLUTIONS TO SELECTED EXERCISES253

SUGGESTIONS FOR FURTHER STUDY267

INDEX269

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