Partial Differential Equations

by John, Fritz

• ISBN13: 9780387906096
• ISBN10: 0387906096
• Edition: 4th
• Format: Hardcover
• Copyright: 1982-03-01
• Publisher: Springer Nature

Summary

Partial Differential Equations is a very well accepted introduction to the subject. In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis. Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy's example of a linear equation without solutions.

Table of Contents

The Single First-Order Equation	1	(33)
Introduction	1	(1)
Examples	2	(2)
Analytic Solution and Approximation Methods in a Simple Example	4	(5)
Problems	8	(1)
Quasi-linear Equations	9	(2)
The Cauchy Problem for the Quasi-linear Equation	11	(4)
Examples	15	(4)
Problems	18	(1)
The General First-Order Equation for a Function of Two Variables	19	(5)
The Cauchy Problem	24	(5)
Solutions Generated as Envelopes	29	(4)
Problems	31	(2)
Second-Order Equations: Hyperbolic Equations for Functions of Two Independent Variables	33	(21)
Characteristics for Linear and Quasi-linear Second-order Equations	33	(2)
Propagation of Singularities	35	(2)
The Linear Second-Order Equation	37	(3)
Problems	39	(1)
The One-Dimensional Wave Equation	40	(6)
Problems	45	(1)
Systems of First-Order Equations	46	(6)
A Quasi-linear System and Simple Waves	52	(2)
Problem	53	(1)
Characteristic Manifolds and the Cauchy Problem	54	(40)
Notation of Laurent Schwartz	54	(2)
Problems	55	(1)
The Cauchy Problem	56	(5)
Problems	61	(1)
Real Analytic Functions and the Cauchy-Kowalevski Theorem	61	(18)
Multiple Infinite series	62	(1)
Problems	63	(1)
Real analytic functions	64	(5)
Problems	69	(1)
Analytic and real analytic functions	70	(2)
Problems	72	(1)
The Proof of the Cauchy-Kowalevski theorem	73	(5)
Problems	78	(1)
The Lagrange Green Identity	79	(1)
The Uniqueness Theorem of Holmgren	80	(9)
Problems	88	(1)
Distribution Solutions	89	(5)
Problems	92	(2)
The Laplace Equation	94	(32)
Green's Identity, Fundamental Solutions, and Poisson's Equation	94	(9)
Problems	101	(2)
The Maximum Principle	103	(3)
Problems	105	(1)
The Dirichlet Problem, Green's Function, and Poisson's Formula	106	(5)
Problems	110	(1)
Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions (``Perron's Method'')	111	(6)
Problems	116	(1)
Solution of the Dirichlet Problem by Hilbert-Space Methods	117	(9)
Problems	125	(1)
Hyperbolic Equations in Higher Dimensions	126	(59)
The Wave Equation in n-Dimensional Space	126	(17)
The method of spherical means	126	(6)
Problems	132	(1)
Hadamard's method of descent	133	(1)
Problems	134	(1)
Duhamel's principle and the general Cauchy problem	135	(4)
Problem	139	(1)
Initial-boundary-value problems (``Mixed'' problems)	139	(3)
Problems	142	(1)
Higher-Order Hyperbolic Equations with Constant Coefficients	143	(20)
Standard form of the initial-value problem	143	(2)
Problem	145	(1)
Solution by Fourier transformation	145	(11)
Problems	156	(1)
Solution of a mixed problem by Fourier transformation	157	(1)
The method of plane waves	158	(3)
Problems	161	(2)
Symmetric Hyperbolic Systems	163	(22)
The basic energy inequality	163	(6)
Problems	169	(3)
Existence of solutions by the method of finite differences	172	(9)
Problems	181	(1)
Existence of solutions by the method of approximation by analytic functions (Method of Schauder)	182	(3)
Higher-Order Elliptic Equations with Constant Coefficients	185	(21)
The Fundamental Solution for Odd n	186	(4)
Problems	188	(2)
The Dirichlet Problem	190	(8)
Problems	195	(3)
More on the Hilbert Space Hμ0 and the Assumption of Boundary Values in the Dirichlet Problem	198	(8)
Problems	201	(5)
Parabolic Equations	206	(29)
The Heat Equation	206	(21)
The initial-value problem	206	(7)
Problems	213	(2)
Maximum principle, uniqueness, and regularity	215	(5)
Problem	220	(1)
A mixed problem	220	(1)
Problems	221	(1)
Non-negative solutions	222	(4)
Problems	226	(1)
The Initial-Value Problem for General Second-Order Linear Parabolic Equations	227	(8)
The method of finite differences and the maximum principle	227	(4)
Existence of solutions of the initial-value problem	231	(2)
Problems	233	(2)
H. Lewy's Example of a Linear Equation without Solutions	235	(6)
Problems	239	(2)
Bibliography	241	(2)
Glossary	243	(2)
Index	245