On a problem of determining the right-hand side of the partial integro-differential equation

As it is known, in the inverse problem, apart from the sought-for “basic” solution of the problem (i. e., the solution of the direct problem), the components of the direct problem are unknown. It is required to find these unknown components, so they will be also included in the solution of the inverse problem. To determine these components in the inverse problem, some additional information on the solution of the direct problem is added to the given equations. The additional information is called the inverse problem data. In the proposed article, the specific fourth-order partial integro-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 right-hand side of the equation contains $n$ unknown functions: $\varphi_{i}(t)$, $i = 1,2,\dots,n$. To determine these unknown functions: $\varphi_{i}(t)$, $i = 1,2,\dots,n$ in the inverse problem there is additional information on the solution of the direct problem, i.e., the values of the sought-for “basic” solution to the problem in the inner segments of the investigated region are known, i. e., $u(t,x_{i}) = g_{i}(t)$, $t\in [0,T]$, $x_{i}\in(0,1)$, $i = 1, 2,\dots, n$. 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 integro-differential equation have been found. When solving the inverse problem, the methods of transformations, Green's function, solutions of systems of linear Volterra integral equations have been used. As a result, the inverse problem has been reduced to a system of $(n+1)$ linear Volterra integral equations of the second kind, the solution of which for small $0<T$ exists and is unique. The considered inverse problem can be called the inverse source problem.