Inverse problem for non-homogeneous integro-differential equation of hyperbolic type

Abstract

An inverse problem is considered, which consists in finding a solution and a one-dimensional kernel of the integral term of an inhomogeneous integro-differential equation of hyperbolic type from the conditions that make up the direct problem and some additional condition. First, the direct problem is investigated, while the kernel of the integral term is assumed to be known. By integrating over the characteristics, the given intego-differential equation is reduced to a Volterra integral equation of the second kind and is solved by the method of successive approximations. Further, using additional information about the solution of the direct problem, we obtain an integral equation with respect to the kernel of the integral k(t), of the integral term. Thus, the problem is reduced to solving a system of integral equations of the Volterra type of the second kind. The resulting system is written as an operator equation. To prove the global, unique solvability of this problem, the method of contraction mappings in the space of continuous functions with weighted norms is used. And also the theorem of conditional stability of the solution of the inverse problem is proved, while the method of estimating integrals and Gronoullo’s inequality is used.