AN INITIAL-BOUNDARY VALUE PROBLEM FOR A HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATION

Аннотация

The initial-boundary value problem for higher-order partial differential equations is considered. We study the existence of its classical solutions, and also propose a method for finding approximate solutions. Paper establishes sufficient conditions for the existence and uniqueness of the classical solution of the problem under consideration. Introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for second-order hyperbolic equations with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for higher-order partial differential equations are obtained in terms of the initial data. Solvability of the initial-boundary value problem for higher-order partial differential equations is connected with solvability of the nonlocal problem for second-order hyperbolic equations.