ON DETERMINATION OF THE COEFFICIENT AND KERNEL IN AN INTEGRO -DIFFERENTIAL EQUATION OF PARABOLIC TYPE

The inverse problem of determination of x-dependent coefficient a(x) at u and the kernel k(t) functions in the one-dimensional integro–differential parabolic equation is investigated. The direct problem is the initial-boundary problem for this equation. Firstly, we studied the solvability of the direct problem, by used to the Fourier method and approximation series methods. As additional information for solving inverse problem, the solution of the direct problem by over determining condition is given. The problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of inverse problem solution.