Introduction to Partial Differential Equations with Applications

Cover......Page 1
Title page......Page 3
PREFACE......Page 6
TABLE OF CONTENTS......Page 8
1. Sets and functions......Page 12
2. Surfaces and their normals. The implicit function theorem......Page 16
3. Curves and their tangents......Page 22
4. The initial value problem for ordinary differential equations and systems......Page 27
References for Chapter I......Page 34
1. Integral curves of vector fields......Page 35
2. Methods of solution of dx/P = dy/Q = dz/R......Page 46
3. The general solution of PUx + Quu + Ru z = O......Page 52
4. Construction of an integral surface of a vector field containing a given curve......Page 55
5. Applications to plasma physics and to solenoidal vector fields......Page 62
References for Chapter II......Page 67
1. First order partial differential equations......Page 68
2. The general integral of PZx + Qzu = R......Page 70
3. The initial value problem for quasi-linear first order equations. Existence and uniqueness of solution......Page 75
4. The initial value problem for quasi-linear first order equations. Nonexistence and nonuniqueness of solutions......Page 80
5. The initial value problem for conservation laws. The development of shocks......Page 83
6. Applications to problems in traffic flow and gas dynamics......Page 87
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 97
References for Chapter III......Page 106
1. Taylor series. Analytic functions......Page 107
2. The Cauchy- Kovalevsky theorem......Page 111
References for Chapter IV......Page 122
1. Linear partial differential operators and their characteristic curves and surfaces......Page 123
2. Methods for finding characteristic curves and surfaces. Examples......Page 128
3. The importance of characteristics. A very simple example......Page 135
4. The initial value problem for linear first order equations in two independent variables......Page 137
5. The general Cauchy problem. The Cauchy- Kovalevsky theorem and Holmgren\'s uniqueness theorem......Page 143
6. Canonical form of first order equations......Page 144
7. Classification and canonical forms of second order equations in two independent variables......Page 148
8. Second order equations in two or more independent variables......Page 154
9. The principle of superposition......Page 160
Reference for Chapter V......Page 163
1. The divergence theorem and the Green\'s identities......Page 164
2. The equation of heat conduction......Page 167
3. Laplace\'s equation......Page 172
4. The wave equation......Page 173
5. Well-posed problems......Page 177
Reference for Chapter VI......Page 181
CHAPTER VII. LAPLACE\'S EQUATION......Page 182
1. Harmonic functions......Page 183
2. Some elementary harmonic functions. The method of separation of variables......Page 184
3. Changes of variables yielding new harmonic functions. Inversion with respect to circles and spheres......Page 190
4. Boundary value problems associated with Laplace\'s equation......Page 197
5. A representation theorem. The mean value property and the maximum principle for harmonic functions......Page 202
6. The well-posedness of the Dirichlet problem......Page 208
7. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson\'s integral......Page 210
8. Introduction to Fourier series......Page 217
9. Solution of the Dirichlet problem using Green\'s functions......Page 234
10. The Green\'s function and the solution to the Dirichlet problem for a ball in R3......Page 237
11. Further properties of harmonic functions......Page 243
12. The Dirichlet problem in unbounded domains......Page 246
13. Determination of the Green\'s function by the method of electrostatic images......Page 251
14. Analytic functions of a complex variable and Laplace\'s equation in two dimensions......Page 256
15. The method of finite differences......Page 260
16. The Neumann problem......Page 268
References for Chapter VII......Page 271
CHAPTER VIII. THE WAVE EQUATION......Page 272
1. Some solutions of the wave equation. Plane and spherical waves......Page 273
2. The initial value problem......Page 282
3. The domain of dependence inequality. The energy method......Page 285
4. Uniqueness in the initial value problem. Domain of dependence and range of influence. Conservation of energy......Page 291
5. Solution of the initial value problem. Kirchhoff\'s formula. The method of descent......Page 295
6. Discussion of the solution of the initial value problem. Huygens\' principle. Diffusion of waves......Page 302
7. Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Reflection of waves......Page 310
8. The vibrating string......Page 319
9. Vibrations of a rectangular membrane......Page 328
10. Vibrations in finite regions. The general method of separation of variables and eigenfunction expansions. Vibrations of a circular membrane......Page 333
References for Chapter VIII......Page 341
1. Heat conduction in a finite rod. The maximum-minimum principle and its consequences......Page 342
2. Solution of the initial-boundary value problem for the one-dimensional heat equation......Page 347
3. The initial value problem for the one-dimensional heat equation......Page 354
4. Heat conduction in more than one space dimension......Page 360
5. An application to transistor theory......Page 364
References for Chapter IX......Page 367
1. Examples of systems. Matrix notation......Page 368
2. Linear hyperbolic systems. Reduction to canonical form......Page 372
3. The method of characteristics for linear hyperbolic systems. Application to electrical transmission lines......Page 378
4. Quasi-linear hyperbolic systems......Page 391
5. One-dimensional isentropic flow of an inviscid gas. Simple waves......Page 392
References for Chapter X......Page 402
GUIDE TO FURTHER STUDY......Page 403
BIBLIOGRAPHY FOR FURTHER STUDY......Page 406
ANSWERS TO SELECTED PROBLEMS......Page 408
INDEX......Page 412
Back cover......Page 417