Colton D. Solution of boundary value problems by the method of integral operators

London: Pitman Publishing, 1976. — 148 p. — ISBN: 0-273-00307-0.These lecture notes are intended to be a companion volume to D. Colton, Partial Differential Equations in the Complex Domain, Pitman Press Lecture Note Series, London, 1976.
In that book a general survey was given of the analytic theory of partial differential equations, with particular emphasis on improperly posed initial value problems and the analytic continuation of solutions to partial differential equations.
The use of integral operators to solve boundary value and initial-boundary value problems arising in mathematical physics was discussed only briefly. In the present set of notes this topic now becomes the main theme, and the interplay between analytic continuation and the approximation of solutions to partial differential equations is developed in some detail.
With the idea that these two sets of lectures should be read together, we have minimized overlapping topics, while at the same time keeping each set of lectures self-contained.
Indeed the only topics common to "Partial Differential Equations in the Complex Domain" and the present volume are integral operators for elliptic equations in two independent variables (which is treated in considerably more detail in the present set of notes) and the inverse Stefan problem for the heat equation in one space variable (which occupy only a few pages in both the previous and the present volume).
The present set of lectures was given during the academic year 1974-75 while the author was a Guest Professor at the University of Konstanz.
The prerequisites for the course were a one semester course in partial differential equations and a one semester course in analytic function theory.Elliptic Equations in Two Independent Variables
Analytic Continuation
Integral Operators
Complete Families of Solutions
The Bergman Kernel Function
Inverse Methods in Compressible Fluid Flow
Parabolic Equations in One Space Variable
Integral Operators
Reflection Principles
Initial-Boundary Value Problems
Parabolic Equations in Two Space Variables
Integral Operators and the Riemann Function
Complete Families of Solutions
The Inverse Stefan Problem
The Method of Ascent for Elliptic Equations
Interior Domains
Exterior Domains
The Inverse Scattering Problem
Appendix. A Numerical Example
References