Solvability of One Class of Inverse Problem for Partial Differential Equations

In the proposed article, the specific fourth-order partial differential equation with the known initial and boundary conditions is considered. For simplicity, the homogeneous boundary conditions have been examined, since with the help of a linear transformation, the always inhomogeneous boundary conditions can be reduced to the homogeneous ones. The problem is investigated in a rectangle located in the first quarter of the Cartesian coordinate system. To solve the inverse problem, an algorithm has been elaborated and sufficient conditions for the existence and the uniqueness of the solution of the inverse problem for the restoration of the right-hand side in a fourth-order partial differential equation have been found. As a result, the inverse problem has been reduced to a system of linear Volterra integral equations of the second kind, the solution of which for small \(0<T\) exists and is unique.