Introduction to Partial Differential Equations with Applications

Cover......Page 1
Title page......Page 2
PREFACE......Page 4
TABLE OF CONTENTS......Page 6
1. Sets and functions......Page 10
2. Surfaces and their normals. The implicit function theorem......Page 14
3. Curves and their tangents......Page 20
4. The initial value problem for ordinary differential equations and systems......Page 25
References for Chapter I......Page 32
1. Integral curves of vector fields......Page 33
2. Methods of solution of dx/P = dy/Q = dz/R......Page 44
3. The general solution of PUx + Quu + Ru z = O......Page 50
4. Construction of an integral surface of a vector field containing a given curve......Page 53
5. Applications to plasma physics and to solenoidal vector fields......Page 60
References for Chapter II......Page 65
1. First order partial differential equations......Page 66
2. The general integral of PZx + Qzu = R......Page 68
3. The initial value problem for quasi-linear first order equations. Existence and uniqueness of solution......Page 73
4. The initial value problem for quasi-linear first order equations. Nonexistence and nonuniqueness of solutions......Page 78
5. The initial value problem for conservation laws. The development of shocks......Page 81
6. Applications to problems in traffic flow and gas dynamics......Page 85
7. The method of probability generating functions. Applications to a trunking problem in a telephone network and to the control of a tropical disease......Page 95
References for Chapter III......Page 104
1. Taylor series. Analytic functions......Page 105
2. The Cauchy- Kovalevsky theorem......Page 109
References for Chapter IV......Page 120
1. Linear partial differential operators and their characteristic curves and surfaces......Page 121
2. Methods for finding characteristic curves and surfaces. Examples......Page 126
3. The importance of characteristics. A very simple example......Page 133
4. The initial value problem for linear first order equations in two independent variables......Page 135
5. The general Cauchy problem. The Cauchy- Kovalevsky theorem and Holmgren's uniqueness theorem......Page 141
6. Canonical form of first order equations......Page 142
7. Classification and canonical forms of second order equations in two independent variables......Page 146
8. Second order equations in two or more independent variables......Page 152
9. The principle of superposition......Page 158
Reference for Chapter V......Page 161
1. The divergence theorem and the Green's identities......Page 162
2. The equation of heat conduction......Page 165
3. Laplace's equation......Page 170
4. The wave equation......Page 171
5. Well-posed problems......Page 175
Reference for Chapter VI......Page 179
CHAPTER VII. LAPLACE'S EQUATION......Page 180
1. Harmonic functions......Page 181
2. Some elementary harmonic functions. The method of separation of variables......Page 182
3. Changes of variables yielding new harmonic functions. Inversion with respect to circles and spheres......Page 188
4. Boundary value problems associated with Laplace's equation......Page 195
5. A representation theorem. The mean value property and the maximum principle for harmonic functions......Page 200
6. The well-posedness of the Dirichlet problem......Page 206
7. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral......Page 208
8. Introduction to Fourier series......Page 215
9. Solution of the Dirichlet problem using Green's functions......Page 232
10. The Green's function and the solution to the Dirichlet problem for a ball in R3......Page 235
11. Further properties of harmonic functions......Page 241
12. The Dirichlet problem in unbounded domains......Page 244
13. Determination of the Green's function by the method of electrostatic images......Page 249
14. Analytic functions of a complex variable and Laplace's equation in two dimensions......Page 254
15. The method of finite differences......Page 258
16. The Neumann problem......Page 266
References for Chapter VII......Page 269
CHAPTER VIII. THE WAVE EQUATION......Page 270
1. Some solutions of the wave equation. Plane and spherical waves......Page 271
2. The initial value problem......Page 280
3. The domain of dependence inequality. The energy method......Page 283
4. Uniqueness in the initial value problem. Domain of dependence and range of influence. Conservation of energy......Page 289
5. Solution of the initial value problem. Kirchhoff's formula. The method of descent......Page 293
6. Discussion of the solution of the initial value problem. Huygens' principle. Diffusion of waves......Page 300
7. Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Reflection of waves......Page 308
8. The vibrating string......Page 317
9. Vibrations of a rectangular membrane......Page 326
10. Vibrations in finite regions. The general method of separation of variables and eigenfunction expansions. Vibrations of a circular membrane......Page 331
References for Chapter VIII......Page 339
1. Heat conduction in a finite rod. The maximum-minimum principle and its consequences......Page 340
2. Solution of the initial-boundary value problem for the one-dimensional heat equation......Page 345
3. The initial value problem for the one-dimensional heat equation......Page 352
4. Heat conduction in more than one space dimension......Page 358
5. An application to transistor theory......Page 362
References for Chapter IX......Page 365
1. Examples of systems. Matrix notation......Page 366
2. Linear hyperbolic systems. Reduction to canonical form......Page 370
3. The method of characteristics for linear hyperbolic systems. Application to electrical transmission lines......Page 376
4. Quasi-linear hyperbolic systems......Page 389
5. One-dimensional isentropic flow of an inviscid gas. Simple waves......Page 390
References for Chapter X......Page 400
GUIDE TO FURTHER STUDY......Page 401
BIBLIOGRAPHY FOR FURTHER STUDY......Page 404
ANSWERS TO SELECTED PROBLEMS......Page 406
INDEX......Page 410