The Dirichlet problem in an infinite layer for a system of differential equations with shifts

Abstract

In this paper, we study the problem with data on the boundary of the infinite layer { ( t , x ) : t ∈ ( 0 , h ) , x ∈ R s } , h > 0 , s ∈ N , \{(t,x):t\in(0,h),\,x\in\mathbb{R}^{s}\},\quad h>0,\,s\in\mathbb{N}, for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables x 1 , x 2 , … , x s x_{1},x_{2},\ldots,x_{s} . We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.