On determination of minor coefficient in a parabolic equation of the second order

E. I. Safonov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 4, pp. 30-40

Résumé

An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considered. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this time-dependent coefficient using data of the initial-boundary value problem and point conditions of overdetermination. Cases of the Dirichlet boundary conditions and oblique derivative conditions are considered. Conditions under which the theorem of existence and solution uniqueness is applicable for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element method and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geometry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the finite difference method. Results of numerical experiments are presented, and numerical solution of the model inverse problem is constructed in the case of Neumann boundary conditions; dependency of an error in calculation of the controlling parameter on the variation of the equation coefficients and the noise level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations show a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, though the graph of recovered coefficient remains close to the solution graph and repeats its outlines.

Détail

Keywords: finite element method, inverse problem.
Mots-clés : parabolic equation

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