Asymptotic solutions of scalar integro-differential equations with partial derivatives and with rapidly oscillating coefficients

Abstract

The work is devoted to the development of an asymptotic integration algorithm for the Cauchy problem for a singularly perturbed partial differential integro-differential equation with rapidly oscillating coefficients, which describe various physical processes in micro-inhomogeneous media.This direction in the theory of partial differential equations is developing intensively and finds numerous applications in radiophysics, electrical engineering, filtering theory, phase transition theory, elasticity theory, and other branches of physics, mechanics, and technology. For studies of such processes, asymptotic methods are usually used. It is known that currently rapidly developing numerical methods do not exclude asymptotic. This happens for a number of reasons. Firstly, a reasonably constructed asymptotics, especially its main term, carries information that is important for applications about the qualitative behavior of the solution and, in this sense, to some extent replaces the exact solution, which most often cannot be found. Secondly, as follows from the above, knowledge of the solution structure helps in the development of numerical methods for solving complex problems; therefore, the development of asymptotic methods contributes to the development of numerical methods. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.