Elements of Partial Differential Equations

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ISBN: 9780486452975 | 0486452972
Cover: Paperback
Copyright: 8/11/2006

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This text features numerous worked examples in its presentation of elements from the theory of partial differential equations. It emphasizes forms suitable for students and researchers whose interest lies in solving equations rather than in general theory. Solutions to odd-numbered problems appear at the end. 1957 edition.

Preface

iii

CHAPTER 1. ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES

(43)

1. Surfaces and Curves in Three Dimensions

(6)

2. Simultaneous Differential Equations of the First Order and the First Degree in Three Variables

(3)

3. Methods of Solution of dx|P = dy|Q dz|R

(5)

4. Orthogonal Trajectories of a System of Curves on a Surface

(3)

5. Pfaffian Differential Forms and Equations

(8)

6. Solution of Pfaffian Differential Equations in Three Variables

(7)

7. Carathéodory's Theorem

(6)

8. Application to Thermodynamics

(3)

Miscellaneous Problems

(2)

CHAPTER 2. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

(44)

1. Partial Differential Equations

(1)

2. Origins of First-order Partial Differential Equations

(2)

3. Cauchy's Problem for First-order Equations

(2)

4. Linear Equations of the First Order

(7)

5. Integral Surfaces Passing through a Given Curve

(1)

6. Surfaces Orthogonal to a Given System of Surfaces

(2)

7. Nonlinear Partial Differential Equations of the First Order

(2)

8. Cauchy's Method of Characteristics

(6)

9. Compatible Systems of First-order Equations

(2)

10. Charpit's Method

(1)

11. Special Types of First-order Equations

(3)

12. Solutions Satisfying Given Conditions

(5)

13. Jacobi's Method

(3)

14. Applications of First-order Equations

(4)

Miscellaneous Problems

(3)

CHAPTER 3. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

(53)

1. The Origin of Second-order Equations

(2)

2. Second-order Equations in Physics

(4)

3. Higher-order Equations in Physics

(2)

4. Linear Partial Differential Equations with Constant Coefficients

(9)

5. Equations with Variable Coefficients

105

(5)

6. Characteristic Curves of Second-order Equations

110

(5)

7. Characteristics of Equations in Three Variables

115

(4)

8. The Solution of Linear Hyperbolic Equations

119

(4)

9. Separation of Variables

123

(3)

10. The Method of Integral Transforms

126

(5)

11. Nonlinear Equations of the Second Order

131

(5)

Miscellaneous Problems

136

(5)

CHAPTER 4. LAPLACE'S EQUATION

141

(68)

1. The Occurrence of Laplace's Equation in Physics

141

(4)

2. Elementary Solutions of Laplace's Equation

145

(3)

3. Families of Equipotential Surfaces

148

(3)

4. Boundary Value Problems

151

(5)

5. Separation of Variables

156

(5)

6. Problems with Axial Symmetry

161

(3)

7. Kelvin's Inversion Theorem

164

(3)

8. The Theory of Green's Function for Laplace's Equation

167

(7)

9. The Relation of Dirichlet's Problem to the Calculus of Variations

174

(1)

10. "Mixed" Boundary Value Problems

175

(5)

11. The Two-dimensional Laplace Equation

180

(4)

12. Relation of the Logarithmic Potential to the Theory of Functions

184

(9)

13. Green's Function for the Two-dimensional Equation

193

(4)

Miscellaneous Problems

197

(12)

CHAPTER 5. THE WAVE EQUATION

209

(65)

1. The Occurrence of the Wave Equation in Physics

209

(6)

2. Elementary Solutions of the One-dimensional Wave Equation

215

(6)

3. The Riemann-Volterra Solution of the One-dimensional Wave Equation

221

(5)

4. Vibrating Membranes: Application of the Calculus of Variations

226

(6)

5. Three-dimensional Problems

232

(7)

6. General Solutions of the Wave Equation

239

(5)

7. Green's Function for the Wave Equation

244

(5)

8. The Nonhomogeneous Wave Equation

249

(5)

9. Riesz's Integrals

254

(3)

10. The Propagation of Sound Waves of Finite Amplitude

257

(5)

Miscellaneous Problems

262

(12)

CHAPTER 6. THE DIFFUSION EQUATION

274

(35)

1. The Occurrence of the Diffusion Equation in Physics

274

(4)

2. The Resolution of Boundary Value Problems for the Diffusion Equation

278

(4)

3. Elementary. Solutions of the Diffusion Equation

282

(4)

4. Separation of Variables

286

(4)

5. The Use of Integral Transforms

290

(4)

6. The Use of Green's Functions

294

(5)

7. The Diffusion Equation with Sources

299

(4)

Miscellaneous Problems

303

(6)

APPENDIX. SYSTEMS OF SURFACES

309

(6)

1. One-parameter Systems

309

(2)

2. Two-parameter Systems

311

(1)

3. The Edge of Regression

312

(2)

4. Ruled Surfaces

314

(1)

Solutions to the Odd-numbered Problems

315

(8)

Index

323

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Elements of Partial Differential Equations

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