Integral equations in identification of external force and heat source density dynamics

We consider a linear inhomogeneous wave equation and linear inhomogeneous heat equation with initial and boundary conditions. It is assumed that the inhomogeneous terms describing the external force and heat source in the model are decomposed into Fourier series uniformly convergent together with the derivatives up to the second order. In this case, time-dependent expansion coefficients are to be determined. For the purpose of determination of the unknown coefficients, non-local boundary conditions are introduced in accordance with the averaged dynamics required in the model. The nonlocal condition enables the observation of the averaged dynamics of the process. Sufficient conditions are given for the unique classical solution existence. A method for finding the solution of the problem is proposed by reducing to the system of Volterra integral equations of the first kind, which is explicitly constructed in the work. The solution is constructed in explicit form by reduction to Volterra integral equations of the second kind with kernels that admit the construction of the resolvent by means of the Laplace transform. Thus, the work provides a way to solve the identification problem in an analytical form. An illustrative example demonstrating the effectiveness of the proposed approach is given. The statement of the identification problem and the method for solving it allow generalizations also in the case of a system of inhomogeneous equations. The results can be useful in the formulation and solution of the optimization problems of the boundary control process.