On one approach to solve a nonlocal problem with parameter for a second order partial integro-differential equation of hyperbolic type

Abstract

A linear nonlocal problem with a parameter for partial integro-differential equations of hy-perbolic type is considered. This problem is investigated by the Dzhumabaev parameterization method. We offer an algorithm for solving nonlocal problems with parameter for partial integro-differential equations of hyperbolic type. First, the original problem is reduce to an equivalent problem consisting a family of bound-ary value problems for ordinary integro-differential equations with parameters and integral relations. Then, we reduced the family of boundary value problems for ordinary integro-differential equations with parame-ters to a family of special Cauchy problems for ordinary integro-differential equations with parameters in subdomains and functional relations. At fixed values of parameters the family of special Cauchy problems for ordinary integro-differential equations in subdomains has a unique solution. A system of linear function-al equations with respect to parameters is compiled. We propose an algorithm for finding an approximate solution to the equivalent problem. This algorithm includes the approximate solution of the family of Cauchy problems for ordinary differential equations and solving the linear system of functional equations.