**Authors:** Aliona I. Dreglea, Nikolay A. Sidorov

### Abstract

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.

Dreglea Aliona Ivanovna

Vinogradov Institute of Geochemistry SB RAS

1Рђ Favorsky str., Irkutsk, 664033, Russia

E-mail:

Sidorov Nikolay Alexandrovich

Irkutsk State University

1, K. Marks str., Irkutsk, 664003

E-mail:

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