Identification of the non-stationary coefficient of the lowest derivative in a parabolic equation

Vasil'ev V.I., Kardashevsky A.M., Ivanov D.K., Kardashevskaia K.S.
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This paper presents a non-iterative method for solving an inverse problem for a parabolictype equation with an unknown time-dependent coefficient at the first spatial derivative. The overdetermination condition is specified as a definite integral of the unknown function with a weighting factor over the spatial domain or its subdomain. The study is motivated by the need to identify dynamic parameters in applied problems, particularly in modeling transport processes in biological fluids, where the flow velocity may vary over time. In contrast to conventional iterative methods that require substantial computational effort and careful selection of regularization parameters, an original approach based on solution decomposition is proposed. At each time layer, the solution is represented as a linear combination of solutions to two auxiliary systems with the same matrix and different right-hand sides, followed by the determination of the unknown coefficient from a discrete analogue of the overdetermination condition. This approach eliminates the need for an iterative procedure. In the presence of inexact overdetermination data, the highest reconstruction accuracy is achieved using a quasi-solution. Numerical experiments on test problems demonstrate high accuracy in reconstructing the unknown functions under small perturbations of the overdetermination condition. The results indicate strong potential for applications in medical diagnostics and other fields requiring rapid processing of experimental data.

Keywords: convective diffusion equation, coefficient inverse problem, finite difference method, decomposition method
Citation in English: Vasil'ev V.I., Kardashevsky A.M., Ivanov D.K., Kardashevskaia K.S. Identification of the non-stationary coefficient of the lowest derivative in a parabolic equation // Computer Research and Modeling, 2026, vol. 18, no. 3, pp. 607-620
Citation in English: Vasil'ev V.I., Kardashevsky A.M., Ivanov D.K., Kardashevskaia K.S. Identification of the non-stationary coefficient of the lowest derivative in a parabolic equation // Computer Research and Modeling, 2026, vol. 18, no. 3, pp. 607-620
DOI: 10.20537/2076-7633-2026-18-3-607-620
URL: http://crm-en.ics.org.ru/journal/article/3735/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Vasil'ev V.I., Kardashevsky A.M., Ivanov D.K., Kardashevskaia K.S.

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