Partial differential equations : methods, applications and theories / Harumi Hattori, West Virginia University, USA.

Format:

Book

Author/Creator:

Hattori, Harumi, author.

Language:

English

Subjects (All):

Differential equations, Partial.

Physical Description:

xvi, 375 pages : illustrations ; 24 cm

Place of Publication:

Singapore : World Scientific, [2013]

Summary:

This volume is an introductory level textbook for partial differential equations (PDE's) and suitable for a one-semester undergraduate level or two-semester graduate level course in PDE's or applied mathematics. Chapters One to Five are organized according to the equations and the basic PDE'S are introduced in an easy to understand manner. They include the first-order equations and the three fundamental second-order equations, i.e. the heat, wave and Laplace equations. Through these equations we learn the types of problems, how we pose the problems, and the methods of solutions such as the separation of variables and the method of characteristics. The modeling aspects are explained as well. The methods introduced in earlier chapters are developed further in Chapters Six to Twelve. They include the Fourier series, the Fourier and the Laplace transforms, and the Green's functions. The equations in higher dimensions are also discussed in detail. This volume is application-oriented and rich in examples. Going through these examples, the reader is able to easily grasp the basics of PDE's. Book jacket.

Contents:

1 First and Second Order Linear Equations - Preparation 1

1.1 Terminologies 1

1.2 Linearity 3

1.2.1 Superposition Principle 3

1.2.2 Linear Independence 6

1.3 First Order Linear Equations 8

1.3.1 Initial Value Problems 8

1.3.2 General Solutions 14

1.4 Classification of Second Order Linear Equations 17

1.5 Well-posedness 23

2 Heat Equation 25

2.1 Derivation of the Heat Equation 26

2.1.1 One-dimensional Case 26

2.1.2 Divergence Theorem 28

2.1.3 Multi-dimensional Case 31

2.2 Initial Boundary Value Problems 33

2.3 Homogeneous Boundary Conditions 35

2.3.1 Temperature is Fixed at Zero at Both Ends 35

2.3.2 Brief Discussion of the Fourier Series 40

2.3.3 Both Ends are Insulated 43

2.3.4 Temperature of One End is Zero and the Other End is Insulated 46

2.4 Non-homogeneous Boundary Conditions 51

2.4.1 Steady State Solutions 51

2.4.2 Non-homogeneous Boundary Conditions 54

2.5 Robin Boundary Conditions 57

2.6 Infinite Domain Problems 62

2.6.1 Initial Value Problems 62

2.6.2 Initial Value Problems via Fourier Transform 65

2.6.3 Semi-infinite Domains 69

2.7 Maximum Principle, Energy Method, and Uniquness of Solutions 72

2.7.1 Maximum Principles 72

2.7.2 Energy Method 75

3 Wave Equation 79

3.1 Derivation of Wave Equation 79

3.1.1 One-dimensional Case 79

3.1.2 Multi-dimensional Case 81

3.2 Initial Value Problems 82

3.2.1 Homogeneous Wave Equation 82

3.2.2 Non-homogeneous Wave Equation 91

3.3 Wave Reflection Problems 95

3.3.1 Homogeneous Boundary Conditions 95

3.3.2 Non-homogeneous Boundary Conditions 108

3.4 Initial Boundary Value Problems 111

3.5 Energy Method 115

4 Laplace Equation 119

4.1 Motivations 119

4.2 Boundary Value Problems - Separation of Variables 121

4.2.1 Laplace Equation on a Rectangular Domain 121

4.2.2 Laplace Equation on a Circular Disk 125

4.3 Fundamental Solution 131

4.3.1 Green's Identity 132

4.3.2 Derivation of Fundamental Solution 133

4.3.3 Green's Identity and Fundamental Solution 133

4.4 Green's Function 136

4.4.1 Definition 136

4.4.2 Green's Function for a Half Space 137

4.4.3 Green's Function for a Ball 138

4.4.4 Symmetry of Green's Function 141

4.5 Properties of Harmonic Functions 142

4.5.1 Mean Value Property 143

4.5.2 The Maximum Principle and Uniqueness 144

4.6 Well-posedness Issues 146

4.6.1 Laplace Equation 146

4.6.2 Wave Equation 148

5 First Order Equations Revisited 151

5.1 First Order Quasilinear Equations 151

5.2 An Application of Quasilinear Equations 157

5.2.1 Scalar Conservation Law 157

5.2.2 Rankine-Hugonio Condition 161

5.2.3 Weak Solutions 164

5.2.4 Entropy Condition and Admissibility Criterion 166

5.2.5 Traffic Flow Problem 170

5.3 First Order Nonlinear Equations 175

5.4 An Application of Nonlinear Equations - Optimal Control Problem 178

5.5 Systems of First Order Equations 182

5.5.1 2 × 2 System 182

5.5.2 n × n System 185

6 Fourier Series and Eigenvalue Problems 187

6.1 Even, Odd, and Periodic Functions 187

6.1.1 Even and Odd Functions 187

6.1.2 Periodic Functions 189

6.2 Fourier Series 190

6.2.1 Fourier Series 190

6.2.2 Fourier Sine and Cosine Series 193

6.3 Fourier Convergence Theorems 195

6.3.1 Mean-square Convergence 196

6.3.2 Pointwise Convergence 199

6.3.3 Uniform Convergence 205

6.4 Derivatives of Fourier Series 209

6.5 Eigenvalue Problems 214

6.5.1 The Sturm-Liouville Problems 214

6.5.2 Proofs 216

7 Separation of Variables in Higher Dimensions 221

7.1 Rectangular Domains 221

7.2 Eigenvalue Problems 228

7.2.1 Multidimensional Case 228

7.2.2 Gram-Schmidt Orthogonalization Procedure 229

7.2.3 Rayleigh Quotient 230

7.3 Eigenfunction Expansions 234

7.3.1 Non-homogeneous Boundary Conditions 234

7.3.2 Homogeneous Boundary Conditions 237

7.3.3 Hybrid Method 240

8 More Separation of Variables 245

8.1 Circular Domains 245

8.1.1 Initial Boundary Value Problems 245

8.1.2 Bessel and Modified Bessel Functions 250

8.2 Cylindrical Domains 253

8.2.1 Initial Boundary Value Problems 254

8.2.2 Laplace Equation 257

8.3 Spherical Domains 261

8.3.1 Initial Boundary Value Problems 262

8.3.2 Legendre Equation 266

8.3.3 Laplace Equation 267

9 Fourier Transform 269

9.1 Delta Functions 269

9.1.1 Classical Introduction 269

9.1.2 Modern Introduction 271

9.2 Fourier Transform 275

9.2.1 Complex Form of the Fourier Series 275

9.2.2 Fourier Transform and Inverse 276

9.3 Properties of Fourier Transform 278

9.3.1 Fourier Transform of Derivatives 278

9.3.2 Convolution 279

9.3.3 Plancerel Formula 280

9.4 Applications of Fourier Transform 281

9.4.1 Heat Equation 281

9.4.2 Wave Equation 283

9.4.3 Laplace Equation in a Half Space 284

9.4.4 Black-Scholes-Merton Equation 286

10 Laplace Transform 291

10.1 Laplace Transform and the Inverse 291

10.1.1 Laplace Transform 291

10.1.2 Inverse Transform 293

10.2 Properties of the Laplace Transform 295

10.2.1 Laplace Transform of Derivatives 295

10.2.2 Convolution Theorem 296

10.2.3 Relation with the Fourier Transform 298

10.3 Applications to Differential Equations 300

10.3.1 Applications to ODE's 300

10.3.2 Applications to PDE's 301

11 Higher Dimensional Problems - Other Approaches 307

11.1 Spherical Means and Method of Descent 307

11.1.1 Method of Spherical Means 307

11.1.2 The Method of Descent 311

11.2 Duhamel's Principle 313

11.2.1 Heat Equation 313

11.2.2 Wave Equation 315

12 Green's Functions 317

12.1 Green's Functions for the Laplace Equation 317

12.1.1 Eigenfunction Expansion 317

12.1.2 Modified Green's Function 321

12.2 Green's Functions for the Heat Equation 326

12.2.1 Initial Boundary Value Problems 326

12.2.2 Initial Value Problems 329

12.3 Green's Functions for the Wave Equation 331

12.3.1 Initial Boundary Value Problems 331

12.3.2 Initial Value Problems 334.

Notes:

Includes bibliographical references and index.

ISBN:

9814407569

9789814407564

OCLC:

826660075

Publisher Number:

99953533339