An introduction to partial differential equations / Yehuda Pinchover and Jacob Rubinstein.

Format:

Book

Author/Creator:

Pinchover, Yehuda, 1953-

Contributor:

Rubinstein, Jacob, 1955-

Language:

English

Subjects (All):

Differential equations, Partial--Textbooks.

Differential equations, Partial.

Genre:

Textbooks.

Physical Description:

xii, 371 pages : illustrations ; 26 cm

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2005.

Summary:

A complete introduction to partial differential equations, this textbook provides a rigorous yet accessible guide to students in mathematics, physics, and engineering. The presentation is lively and up to date, putting particular emphasis into developing an appreciation of underlying mathematical theory.

Beginning with basic definitions, properties, and derivations of some fundamental equations from mathematical physics, the book studies first-order equations, classification of second-order equations, and the one-dimensional wave equation. Two chapters are devoted to the separation of variables, while others concentrate on a wide range of topics including elliptic theory, Green's functions, variational, and numerical methods.

A rich collection of worked examples and exercises accompanies the text, along with a large number of illustrations and graphs to provide insight into the numerical examples. Solutions to selected exercises are included for students, while extended solution sets are available to lecturers from www.cambridge.org/052161323X.

Contents:

1.2 Classification 3

1.3 Differential operators and the superposition principle 3

1.4 Differential equations as mathematical models 4

1.5 Associated conditions 17

1.6 Simple examples 20

2 First-order equations 23

2.2 Quasilinear equations 24

2.3 The method of characteristics 25

2.4 Examples of the characteristics method 30

2.5 The existence and uniqueness theorem 36

2.6 The Lagrange method 39

2.7 Conservation laws and shock waves 41

2.8 The eikonal equation 50

2.9 General nonlinear equations 52

3 Second-order linear equations in two indenpendent variables 64

3.2 Classification 64

3.3 Canonical form of hyperbolic equations 67

3.4 Canonical form of parabolic equations 69

3.5 Canonical form of elliptic equations 70

4 The one-dimensional wave equation 76

4.2 Canonical form and general solution 76

4.3 The Cauchy problem and d' Alembert's formula 78

4.4 Domain of dependence and region of influence 82

4.5 The Cauchy problem for the nonhomogeneous wave equation 87

5 The method of separation of variables 98

5.2 Heat equation: homogeneous boundary condition 99

5.3 Separation of variables for the wave equation 109

5.4 Separation of variables for nonhomogeneous equations 114

5.5 The energy method and uniqueness 116

5.6 Further applications of the heat equation 119

6 Sturm-Liouville problems and eigenfunction expansions 130

6.2 The Sturm-Liouville problem 133

6.3 Inner product spaces and orthonormal systems 136

6.4 The basic properties of Sturm-Liouville eigenfunctions and eigenvalues 141

6.5 Nonhomogeneous equations 159

6.6 Nonhomogeneous boundary conditions 164

7 Elliptic equations 173

7.2 Basic properties of elliptic problems 173

7.3 The maximum principle 178

7.4 Applications of the maximum principle 181

7.5 Green's identities 182

7.6 The maximum principle for the heat equation 184

7.7 Separation of variables for elliptic problems 187

7.8 Poisson's formula 201

8 Green's functions and integral representations 208

8.2 Green's function for Dirichlet problem in the plane 209

8.3 Neumann's function in the plane 219

8.4 The heat kernel 221

9 Equations in high dimensions 226

9.2 First-order equations 226

9.3 Classification of second-order equations 228

9.4 The wave equation in R[superscript 2] and R[superscript 3] 234

9.5 The eigenvalue problem for the Laplace equation 242

9.6 Separation of variables for the heat equation 258

9.7 Separation of variables for the wave equation 259

9.8 Separation of variables for the Laplace equation 261

9.9 Schrodinger equation for the hydrogen atom 263

9.10 Musical instruments 266

9.11 Green's functions in higher dimensions 269

9.12 Heat kernel in higher dimensions 275

10 Variational methods 282

10.1 Calculus of variations 282

10.2 Function spaces and weak formulation 296

11 Numerical methods 309

11.2 Finite differences 311

11.3 The heat equation: explicit and implicit schemes, stability, consistency and convergence 312

11.4 Laplace equation 318

11.5 The wave equation 322

11.6 Numerical solutions of large linear algebraic systems 324

11.7 The finite elements method 329

12 Solutions of odd-numbered problems 337

A.1 Trigonometric formulas 361

A.2 Integration formulas 362

A.3 Elementary ODEs 362

A.4 Differential operators in polar coordinates 363

A.5 Differential operators in spherical coordinates 363.

Notes:

Includes bibliographical references (pages 364-365) and index.

ISBN:

0521848865

052161323X

OCLC:

59223853