Bounder solution on a strip to a system of nonlinear hyperbolic equations with mixed derivatives

Abstract

The system of nonlinear hyperbolic equations with mixed derivatives is considered on the strip. Time variable of the unknown function changes on the whole axis, and the spatial variable belongs to a finite interval. A function, the partial derivative with respect to the spatial variable, is denoted as unknown function, and problem of finding a bounded on the strip solution to the origin system is reduced to the problem of finding a bounded on the strip solution to a system of integro - partial differential equations. The whole axes is divided into parts, and additional functional parameters are introduced as the values of unknown function on the initial lines of sub-domains. For the fixed values of functional parameters, the new unknown functions in the sub-domains are defined as the solutions to the Cauchy problems for integro-partial differential equations of the first order. Using the continuity conditions of the solution on the partition lines, the two-sided infinite system of nonlinear Volterra integral equations of the second kind with respect to introduced functional parameters is obtained. Algorithms for finding solutions of problem with functional parameters are proposed. Conditions for the convergence of algorithms, and existence of bounded on the strip solution of the system of nonlinear hyperbolic equations with mixed derivatives are obtained.