Integral operator methods in the theory of wave propagation and heat conduction

Until recently the method of integral operators as initiated by S. Bergman and I. N. Vekua has been restricted to the case of elliptic equations and the investigation of steady state phenomena. In these lectures we survey the recent developments on the use of integral operators to investigate equations associated with evolutionary phenomena, in particular parabolic equations, pseudoparabolic equations, and the reduced wave equation in a stratified medium. The topics discussed are transformation operators for partial differential equations, reflection principles and their application, the propagation of radio waves around the earth, the propagation of acoustic waves in a spherically stratified medium, low frequency approximations to acoustic scattering problems in a spherically stratified medium, heat conduction in two temperatures, inverse problems in the theory of heat conduction, and Runge's theorem for parabolic equations. Open problems are given at the end of each section.