Geodesic
A numerical method for solving the coefficient inverse problem for diffusion-convection-reaction equation
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 67-78 Cet article a éte moissonné depuis la source Math-Net.Ru

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The two inverse problems on the restoration of coefficients for nonstationary one-dimensional diffusion–convection–reaction equation are considered. The first problem is intended to determine the convective transfer coefficient, which depends only on the time in accordance with the integral overdetermination condition. The second problem allows one to obtain the reaction rate coefficient depending on the time according to the integral overdetermination condition. To solve these problems, at first, a discretization of the time derivative is implemented and the explicit-implicit schemes are used to approximate the operators in both problems. For convective transfer operator in the first problem and reaction operator in the second problem, the explicit sheme was used. For the rest of operators in these problems, the implicit sheme was applied. As a result, both problems are reduced to the differential-difference problems with respect to the functions that depend on the spatial variable. For numerical solution of the problems obtained, a non-iterative computational algorithm is proposed. It is based on reducing of the differentialdifference problem to two direct boundary-value problems and to a linear equation with respect to unknown coefficient. The proposed method was used to carry out the numerical experiments for the model problems.
Mots-clés : diffusion–convection–reaction equation, coefficient inverse problem, explicit-implicit schemes.
Keywords: integral overdetermination condition, differential-difference problem 
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     author = {Kh. M. Gamzaev},
     title = {A numerical method for solving the coefficient inverse problem for diffusion-convection-reaction equation},
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     pages = {67--78},
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Kh. M. Gamzaev. A numerical method for solving the coefficient inverse problem for diffusion-convection-reaction equation. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 67-78. http://geodesic.mathdoc.fr/item/VTGU_2017_50_a5/

[1] Anderson D., Tannehill K., Pletcher R., Computational Fluid Mechanics and Heat Transfer, New York, 1984 | MR

[2] Whitham G. B., Linear and Nonlinear Waves, John Wiley Sons Inc., 1974 | MR | Zbl

[3] Paskonov V. M., Polezhaev V. I., Chudov L. A., Numerical modeling of heat and mass transfer processes, Nauka, M., 1984 | MR

[4] Roache P. J., Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, 1976 | MR

[5] Alifanov O. M., Artyukhin E. A., Rumyantsev S. V., Extreme methods for solving ill-posed problems, Nauka, M., 1988

[6] Samarskiy A. A., Vabishchevich P. N., Numerical methods for solving inverse problems of mathematical physics, Publishing house LCI, M., 2009

[7] Kabanikhin S. I., Inverse and ill-posed problems, Siberian Scientific publishers, Novosibirsk, 2009

[8] Ivanchov N. I., Pabyrivska N. V., “On determination of two time-dependent coefficients in a parabolic equation”, Siberian Mathematical Journal, 43:2 (2002), 323–329 | DOI | MR | Zbl

[9] Kamynin V. L., “The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation”, Mathematical Notes, 94:2 (2013), 205–213 | DOI | DOI | MR | Zbl

[10] Kostin A. B., “Recovery of the coefficient of ut in the heat equation from a condition of nonlocal observation in time”, Computational Mathematics and Mathematical Physics, 55:1 (2015), 85–100 | DOI | DOI | MR | Zbl

[11] Kozhanov A. I., “Parabolic equations with unknown time-dependent coefficients”, Computational Mathematics and Mathematical Physics, 57:6 (2017), 956–966 | DOI | MR | Zbl

[12] Liu Yang, Jian-Ning Yu, Zui-Cha Deng, “An inverse problem of identifying the coefficient of parabolic equation”, Applied Mathematical Modelling, 32:10 (2008), 1984–1995 | DOI | MR | Zbl

[13] N. B. Kerimov, M. I. Ismailov, “An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions”, J. Mathematical Analysis and Applications, 396:2 (2012), 546–554 | DOI | MR | Zbl

[14] Engl H. W., Zou J., “A new approach to convergence rates analysis of Tikhonov regularization for parameter identification in heat conduction”, Inverse Problems, 16 (2000), 1907–1923 | DOI | MR | Zbl

[15] Deng Z. C., Qian K., Rao X. B., Yang L., Luo G. W., “An inverse problem of identifying the source coefficient in a degenerate heat equation”, Inverse Problems in Science and Engineering, 23:3 (2015), 498–517 | DOI | MR | Zbl

[16] Dehghan M., Tatari M., “Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions”, Math. Comput. Modell., 44 (2006), 1160–1168 | DOI | MR | Zbl

[17] Vabishchevich P. N., Vasil'eva M. V., “Explicit-implicit schemes for convection-diffusion-reaction problems”, Numerical Analysis and Applications, 5:4 (2012), 297–306 | DOI | MR | Zbl