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Bul. Acad. Ştiinţe Repub. Mold. Mat., 2018, Number 3, Pages 68–77 (Mi basm487)  

This article is cited in 1 scientific paper (total in 1 paper)

Research articles

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

Aliona I. Dregleaa, Nikolay A. Sidorovb

a Vinogradov Institute of Geochemistry SB RAS, 1A Favorsky str., Irkutsk, 664033, Russia
b Irkutsk State University, 1, K. Marks str., Irkutsk, 664003

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.

Keywords and phrases: BVP, IVP, PDE, second-order hyperbolic equation, wave equation, nonlocal boundary conditions, convergence of Fourier series and of inverse transforms, spectrum, resolvent, Laplace transform, Volterra integral equations, integral observations, identification of an external force, ordinary differential equations, continuous dependence and continuation of solutions, heat equation.

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MSC: 34A34, 34A12, 35L10, 35L05, 35K05, 43A50, 44A10, 45D05
Received: 10.07.2018

Citation: Aliona I. Dreglea, Nikolay A. Sidorov, “Integral equations in identification of external force and heat source density dynamics”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2018, no. 3, 68–77

Citation in format AMSBIB
\by Aliona~I.~Dreglea, Nikolay~A.~Sidorov
\paper Integral equations in identification of external force and heat source density dynamics
\jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat.
\yr 2018
\issue 3
\pages 68--77

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  • Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
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