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.
