Boundary value problem for hyperbolic integro-differential equations of mixed type

Abstract

The boundary value problem for a system of hyperbolic integro-differential equations of
mixed type with degenerate kernels is considered on a rectangular domain. This problem is reduced
to a family of boundary value problems for a system of integro-differential equations of mixed type
and integral relations. The system of integro-differential equations of mixed type is transferred to a
system of Fredholm integro-differential equations. For solving the family of boundary value problems
for integro-differential equations Dzhumabaev’s parametrization method is applied. A new concept
of a general solution to a system of integro-differential equations with parameter is developed. The
domain is divided into N subdomains by a temporary variable, the values of a solution at the interior
lines of the subdomains are considered as additional functional parameters, and a system of
integro-differential equations is reduced to a family of special Cauchy problems on the subdomains for
Fredholm integro-differential equation with functional parameters. Using the solutions to these problems,
a new general solutions to a system of Fredholm integro-differential equations with parameter
is introduced and its properties are established. Based on a general solution, boundary conditions,
and the continuity conditions of a solution at the interior lines of the partition, a system of linear
functional equations with respect to parameters is composed. Its coefficients and right-hand sides
are found by solving the family of special Cauchy problems for Fredholm integro-differential equations
on the subdomains. It is shown that the solvability of the family of boundary value problems
for Fredholm integro-differential equations is equivalent to the solvability of the composed system.
Methods for solving boundary value problems are proposed, which are based on the construction and
solving of these systems. Conditions for the existence and uniqueness of a solution to the boundary
value problem for a system of hyperbolic integro-differential equations of mixed type with degenerate
kernels are obtained.