On a non-local problem for system of partial differential equations of hyperbolic type in a specific domain

Abstract

The non-local problem for second order system of partial differential equations of hyperbolic type is studied in the specific domain. For solving this problem we use a functional parametrization method. This method is an extension of  Dzhumabaev’s parametrization method  to a partial differential equations of hyperbolic type. We introduce a parameter-function, expressed as the unknown function's value at the characteristics  within the given domain. This transforms the nonlocal problem into an equivalent parameterized problem, involving the Goursat problem for a system of partial differential equations of hyperbolic type and an additional relation  based on the functional parameter. Subsequently, starting from the additional condition and the consistency condition, we formulate the Cauchy problem for a system of differential equations with respect to the unknown parameter-function. We develop an algorithm for solving the parameterized problem and demonstrate its convergence. Additionally, we derive conditions for the existence and uniqueness of a solution to the parameterized problem. Unique solvability conditions for the nonlocal problem for  second-order system of partial differential equations of hyperbolic type in a specific domain are established in terms of the initial data.