A SOLVABILITY OF A PROBLEM WITH DATA ON THE CHARACTERISTICS FOR A SYSTEM OF LOADED HYPERBOLIC EQUATIONS SECOND ORDE

Abstract

In this paper we consider  a  problem with data on the characteristics for the system of   loaded hyperbolic equations of second order  on a rectangular domain. The questions of the existence and uniqueness of the classical solution of the considered problem, as well as the continuity  dependence of the solution on the initial data, are investigated.  Problem with data on the characteristics for the system of the hyperbolic equations with continuous initial data always has a unique classical solution.  If  a loaded terms appear in the system of equations, then the problem may not be uniquely solvable. Additional requirements for the coefficients of the system allow us to distinguish a class of solvable problems. In this case, the imposed conditions must be verified and consistent with the theory of boundary value problems for the loaded differential equations. It is known, the main method for solving problems with data on the characteristics for the system of loaded hyperbolic equations is the Riemann method. However, its application requires continuous differentiability of coefficients in the partial derivatives first order  of the system of equations. This article proposes a new approach to solving the problem with data on the characteristics for the system of loaded hyperbolic equations second order based on the Dzhumabaev’s parameterization method. By introducing an additional parameter as the value of the desired function at the load point, the problem is reduced to an equivalent problem with data on the characteristics for a system of hyperbolic equations with a parameter. The unknown parameter is determined from the Cauchy problem for the system of ordinary differential equations, and the unknown function is found from the problem with data on the characteristics for the system of hyperbolic equations for the found parameter. An algorithm for finding an approximate solution to the equivalent problem is proposed and its convergence is proved. Conditions for the unique solvability of the problem with data on the characteristics for the system of loaded hyperbolic equations of  second order are established.