Volume 11, no. 1Pages 145 - 151

# A Numerical Method of Solving the Coefficient Inverse Problem for The Nonlinear Equation of Diffusion-Reaction

Kh.M. Gamzaev
We consider two inverse problems for determining the coefficients for a one-dimensional nonlinear diffusion-reaction equation of the Fisher-Kolmogorov-Petrovsky-Piskunov type. The first problem consists in determining the kinetic coefficient for a nonlinear lower term, depending only on the time variable, according to a given integral condition. And the second problem consists in determining the time-dependent diffusion coefficient, again according to a given integral condition.
To solve both problems, the time derivative of the derivative is first sampled. In the first problem, the diffusion term is approximated in time according to the implicit scheme, and the nonlinear minor term in the semi-explicit scheme. And in the second problem, the diffusion term is approximated in time in an explicitly implicit scheme, and the nonlinear minor term is again in a semi-explicit scheme. As a result, both problems reduce to differential-difference problems with respect to functions depending on the spatial variable. For numerical solution of the problems obtained, a non-iterative computational algorithm is proposed, based on reduction of the differential-difference problem to two direct boundary-value problems and a linear equation with respect to the unknown coefficient. On the basis of the proposed numerical method, numerical experiments were performed for model problems.
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Keywords
diffusion-reaction equation; Fisher-Kolmogorov-Petrovsky-Piskunov equation; coefficient inverse problem; integral condition; differential-difference problem; explicitly implicit schemes.
References
1. Fisher R.A. The Wave of Advance of Advantageous Genes. Annals of Eugenics, 1937, no 7, pp. 355-369. DOI: 10.1111/j.1469-1809.1937.tb02153.x
2. Kolmogorov A.N., Petrovsky I.G., Piskunov I.S. [A Study of the Diffusion Equation with Increase in the Amount of Substance, and Its Application to a Biological Problem]. Bulletin of the Moscow State University, Section A, 1937, vol. 1, no. 6, pp. 1-25.
3. Danilov V.G., Maslov V.P., Volosov K.A. Mathematical Modelling of Heat and Mass Transfer Processes, Dordrecht, Kluwer, 1995. DOI: 10.1007/978-94-011-0409-8
4. Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Walter de Gruyter, 2007. DOI: https:10.1515/9783110205794
5. Kabanikhin S.I. Inverse and Ill-Posed Problems. Theory and Applications. De Gruyter, Germany, 2011. DOI: 10.1515/9783110224016
6. Kamynin V.L. The Inverse Problem of Determining the Lower-Order Coefficient in Parabolic Equations with Integral Observation. Mathematical Notes, 2013, vol. 94, no. 2, pp. 205-213. DOI: 10.1134/S0001434613070201
7. Kozhanov A.I. Parabolic Equations with Unknown Time-Dependent Coefficients. Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 6, pp. 956-966. DOI: 10.1134/S0965542517060082
8. Kerimov N.B., Ismailov M.I. An Inverse Coefficient Problem for the Heat Equation in the Case of Nonlocal Boundary Conditions. Journal of Mathematical Analysis and Applications, 2012, vol. 396, no. 2, pp. 546-554. DOI: 10.1016/j.jmaa.2012.06.046