An Introduction To Partial Differential EquationsPDF|Epub|txt|kindle电子书版本下载

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1 Introduction1

1.1 Preliminaries1

1.2 Classification3

1.3 Differential operators and the superposition principle3

1.4 Differential equations as mathematical models4

1.5 Associated conditions17

1.6 Simple examples20

1.7 Exercises21

2 First-order equations23

2.1 Introduction23

2.2 Quasilinear equations24

2.3 The method of characteristics25

2.4 Examples of the characteristics method30

2.5 The existence and uniqueness theorem36

2.6 The Lagrange method39

2.7 Conservation laws and shock waves41

2.8 The eikonal equation50

2.9 General nonlinear equations52

2.10 Exercises58

3 Second-order linear equations in two indenpendent variables64

3.1 Introduction64

3.2 Classification64

3.3 Canonical form of hyperbolic equations67

3.4 Canonical form of parabolic equations69

3.5 Canonical form of elliptic equations70

3.6 Exercises73

4 The one-dimensional wave equation76

4.1 Introduction76

4.2 Canonical form and general solution76

4.3 The Cauchy problem and d'Alembert's formula78

4.4 Domain of dependence and region of influence82

4.5 The Cauchy problem for the nonhomogeneous wave equation87

4.6 Exercises93

5 The method of separation of variables98

5.1 Introduction98

5.2 Heat equation: homogeneous boundary condition99

5.3 Separation of variables for the wave equation109

5.4 Separation of variables for nonhomogeneous equations114

5.5 The energy method and uniqueness116

5.6 Further applications of the heat equation119

5.7 Exercises124

6 Sturm-Liouville problems and eigenfunction expansions130

6.1 Introduction130

6.2 The Sturm-Liouville problem133

6.3 Inner product spaces and orthonormal systems136

6.4 The basic properties of Sturm-Liouville eigenfunctions and eigenvalues141

6.5 Nonhomogeneous equations159

6.6 Nonhomogeneous boundary conditions164

6.7 Exercises168

7 Elliptic equations173

7.1 Introduction173

7.2 Basic properties of elliptic problems173

7.3 The maximum principle178

7.4 Applications of the maximum principle181

7.5 Green's identities182

7.6 The maximum principle for the heat equation184

7.7 Separation of variables for elliptic problems187

7.8 Poisson's formula201

7.9 Exercises204

8 Green's functions and integral representations208

8.1 Introduction208

8.2 Green's function for Dirichlet problem in the plane209

8.3 Neumann's function in the plane219

8.4 The heat kernel221

8.5 Exercises223

9 Equations in high dimensions226

9.1 Introduction226

9.2 First-order equations226

9.3 Classification of second-order equations228

9.4 The wave equation in R2 and R3234

9.5 The eigenvalue problem for the Laplace equation242

9.6 Separation of variables for the heat equation258

9.7 Separation of variables for the wave equation259

9.8 Separation of variables for the Laplace equation261

9.9 Schrodinger equation for the hydrogen atom263

9.10 Musical instruments266

9.11 Green's functions in higher dimensions269

9.12 Heat kernel in higher dimensions275

9.13 Exercises279

10 Variational methods282

10.1 Calculus of variations282

10.2 Function spaces and weak formulation296

10.3 Exercises306

11 Numerical methods309

11.1 Introduction309

11.2 Finite differences311

11.3 The heat equation: explicit and implicit schemes, stability, consistency and convergence312

11.4 Laplace equation318

11.5 The wave equation322

11.6 Numerical solutions of large linear algebraic systems324

11.7 The finite elements method329

11.8 Exercises334

12 Solutions of odd-numbered problems337

A.l Trigonometric formulas361

A.2 Integration formulas362

A.3 Elementary ODEs362

A.4 Differential operators in polar coordinates363

A.5 Differential operators in spherical coordinates363

References364

Index366