فایل معادلات دیفرانسیل مشتقات جزئی  TYN MYINT-U

 منبع بسیاری از سوالات آزمونهای دانشگاه و تستها

 

 

Preface

 

­Chapter 1.

 

­1.1

 

1.2

 

1.3

 

1.4

 

­Chapter 2.

 

­2.1

 

2.2

 

2.3

 

2.4

 

2.5

 

2.6

 

­Chapter 3.

 

­3.1

 

­3.2

 

­Contents

 

­Introduction

 

­Basic Concepts and Definitions

 

Linear Operators

 

Mathematical Problems

 

Superposi tion

 

Exercises for Chapter 1

 

­Mathematical Models

 

­The Classical Equations

 

The Vibrating String

 

The Vibrating Membrane

 

Waves in an Elastic Medium

 

Conduction of Heat in Solids

 

The Gravitational Potential

 

Exercises for Chapter 2

 

­Classification of Second-order Equations

 

­Second-order Equations in Two Independent

 

Varia bles

 

Canonical Forms

 

­vii

 

­XlII

 

­1

 

4

 

6

 

7

 

10

 

­12

 

12

 

14

 

16

 

20

 

21

 

23

 

­25

 

27

 

3.3

 

3.4

 

3.5

 

­Chapter 4.

 

­4.1

 

4.2

 

­Equations with Constant Coefficients

 

General Solution

 

Summary and Further Simplification

 

Exercises for Chapter 3

 

­33

 

38

 

39

 

41

 

­De Cauchy Problem

 

­The Cauchy Problem 43

 

Cauchy-Kowalewsky Theorem. Hadamard Example

 

46

 

­4.3 The Cauchy Problem for Homogeneous

 

Wave Equation 47

 

4.4 Initial-boundary Value Problem 53

 

4.5 The Cauchy Problem for N onhornogeneous

 

Wave Equation 59

 

4.6 Riemanns Method 62

 

Exercises for Chapter 4 68

 

­Chapter 5.

 

5.1

 

5.2

 

5.3

 

5.4

 

5.5

 

5.6

 

5.7

 

5.8

 

5.9

 

5.10

 

5.11

 

5.12

 

5.13

 

5.14

 

­Fourier Series

 

­Piecewise Continuous Functions 72

 

Even and Odd Functions 75

 

Periodic Functions 77

 

Orthogonality 78

 

Fourier Series 79

 

Convergence in the Mean. Completeness 81

 

Examples of Fourier Series 83

 

Cosine and Sine Series 88

 

Complex Fourier Series 92

 

Change of Interval 93

 

Pointwise Convergence of Fourier Series 96

 

Uniform Convergence of Fourier Series 102

 

Differentiation and Integration of Fourier Series 106

 

Double Fourier Series 110

 

Exercises for Chapter 5 112

 

­viii

 

Chapter 6.

 

­6.1

 

6.2

 

6.3

 

­6.4

 

6.5

 

­6.6

 

6.7

 

6.8

 

­Chapter 7.

 

­De Method of Separation of Variables

 

­Separation of Variables

 

The Vibrating String Problem

 

Existence and Uniqueness of Solution of the

 

Vibrating String Problem

 

The Heat Conduction Problem

 

Existence and Uniqueness of Solution

 

of the Heat Conduction Problem

 

The Laplace and Beam Equations

 

Nonhomogeneous Problems

 

Finite Fourier Transforms

 

Exercises for Chapter 6

 

­Eigenvalue Problems

 

­120

 

123

 

­128

 

134

 

­138

 

141

 

145

 

148

 

153

 

­7.1 Sturm-Liouville Systems 161

 

7.2 Eigenfunctions 165

 

7.3 Bessel Functions 170

 

7.4 Singular Sturm-Liouville Systems 177

 

7.5 Legendre Functions 179

 

7.6 Boundary Value Problems for Ordinary

 

Differential Equations and Greens Functions 184

 

7.7 Construction of Greens Function 189

 

7.8 Generalized Greens Function 192

 

7.9 Eigenvalue Problems and Greens Function 193

 

Exercises for Chapter 7 195

 

­Chapter 8.

 

­8.1

 

8.2

 

­Boundary Value Problems

 

­Boundary Value Problems

 

Maximum and Minimum Principles

 

­ix

 

­201

 

203

 

8.3 Uniqueness and Stability Theorems 204

 

8.4 Dirichlet Problem for a Circle 206

 

8.5 Dirichlet Problem for a Circular Annulus 211

 

8.6 Neumann Problem for a Circle 212

 

8.7 Dirichlet Problem for a Rectangle 213

 

8.8 Dirichlet Problem involving Poisson Equation 217

 

8.9 Neumann Problem for a Rectangle 219

 

Exercises for Chapter 8 222

 

­Chapter 9.

 

­9.1

 

9.2

 

9.3

 

9.4

 

9.5

 

9.6

 

9.7

 

9.8

 

­9.9

 

9.10

 

­9.11

 

9.12

 

­Chapter 10.

 

­10.1

 

10.2

 

10.3

 

10.4

 

­Higher Dimensional Problems

 

­Dirichlet Problem for a Cube 230

 

Dirichlet Problem for a Cylinder 232

 

Dirichlet Problem for a Sphere 236

 

Wave and Heat Equations 241

 

Vibrating Membrane 241

 

Heat Flow in a Rectangular Plate 243

 

Waves in Three Dimensions 245

 

Heat Conduction in a Rectangular Parallelepiped

 

247

 

248

 

­The Hydrogen Atom

 

Nonhomogeneous Problems by the Method

 

of Eigenfunctions

 

Forced Vibration of Membrane

 

Time-dependent Boundary Conditions

 

Exercises for Chapter 9

 

­252

 

252

 

254

 

258

 

­Greens Functions

 

­The Delta Function

 

Greens Function

 

Method of Greens Function

 

Dirichlet Problem for the Laplace Operator

 

­265

 

266

 

268

 

270

 

­x

 

10.5 Dirichlet Problem for the Helmholtz Operator 273

 

10.6 Method of Images 274

 

10.7 Eigenfunction Method 278

 

10.8 Higher Dimensional Problems 280

 

10.9 Neumann Problem 283

 

Exercises for Chapter 10 286

 

­Chapter 11.

 

­Integral Transforms

 

­11.1 Fourier Transforms 289

 

11.2 Properties of Fourier Transform 295

 

11.3 Convolution (Fourier Transform) 299

 

11.4 Step Function and Impulse Function

 

(Fourier Transform) 303

 

11.5 Semi-infinite Region 306

 

11.6 Hankel and Mellon Transforms 307

 

11.7 Laplace Transforms 308

 

11.8 Properties of Laplace Transforms 310

 

11.9 Convolution (Laplace Transform) 314

 

11.10 Step Function and Impulse Function

 

(Laplace transform) 317

 

11.11 Greens Function 324

 

Exercises for Chapter 11 327

 

­Appendix

 

­A.l

 

A.2

 

A.3

 

­Gamma Function

 

Table of Fourier Transforms

 

Table of Laplace Transforms

 

References

 

Answers to Exercises

 

Index

 

­335

 

336

 

337

 

338

 

341

 

363

 

­xi