Partial Differential EquationsIn this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first order and of linear partial differential equations of the second order, using the methods of classical analysis. In spite of the advent of computers and the applications of the methods of functional analysis to the theory of partial differential equations, the classical theory retains its relevance in several important respects. Many branches of classical analysing have their origins in the rigourous discussion of problems in applies mathematics and theoretical physics, and the classical treatment of the theory of partial differential equations still provides the best method of treating many physical problems. A knowledge of the classical theory is essential for pure mathematics who intend to undertake research in this field, whatever approach they ultimately adopt. The numerical analyst needs a knowledge of classical theory in order to decide whether a problem has a unique solution or not. |
Contents
Partial differential equations of the first order | 1 |
Characteristics of equations of the second order | 24 |
Boundary value and initial value problems | 44 |
Equations of hyperbolic type | 54 |
Riemanns method | 77 |
The equation of wave motions | 90 |
Marcel Rieszs method | 107 |
Potential theory in the plane | 131 |
Subharmonic functions and the problem of Dirichlet | 175 |
Equations of elliptic type in the plane | 186 |
Equations of elliptic type in space | 207 |
The equation of heat | 238 |
Appendix | 271 |
Books for further reading | 277 |
| 279 | |
Other editions - View all
Common terms and phrases
absolutely convergent analytic function boundary values bounded domain C₁ Cauchy data Cauchy-Kowalewsky theorem coefficients constant continuous derivatives continuous second derivatives continuously differentiable dx dy dx² dy² elementary solution elliptic type equation of heat equation of wave everywhere exp ikR finite number formula Fourier Green's function Green's theorem harmonic function Hence identically zero infinity initial conditions initial value problem integral surface J²u k²u Laplace's equation linear neighbourhood normal obtain odd function plane Poisson's polar coordinates polynomial problem of Dirichlet prove R₁ reduced wave equation regular arc regular closed curve Riemann-Green function satisfies the equation second order solve spherical supremum tends to zero transformation u(xo u₁ u₂ uniformly unique solution V²u vanishes velocity potential wave equation wave motions yo)² ди др ду дх