Inverse Problem for a Fourth-Order Differential Equation with the Fractional Caputo Operator
Abstract
In this paper we consider an initial-boundary value problem (direct problem) for a fourth order equation with the fractional Caputo derivative. Two inverse problems of determining the right-hand side of the equation by a given solution to the direct problem at some point are studied. The unknown of the first problem is a one-dimensional function depending on a spatial variable, while in the second problem a function depending on a time variable is found. Using eigenvalues and eigenfunctions, a solution to the direct problem is found in the form of Fourier series. Sufficient conditions are established for the given functions, under which the solution to this problem is classical. Using the results obtained for the direct problem and applying the method of integral equations, we study the inverse problems. Thus, the uniqueness and existence theorems of the direct and inverse problems are proved.