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Numerical solution for a nonlinear diffusion model with source terms

  • Temur Jangveladze EMAIL logo , Zurab Kiguradze , Maia Kratsashvili and Beny Neta
Published/Copyright: April 12, 2023
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Abstract

The nonlinear diffusion system with source terms is studied. This model is derived via Maxwell equations describing electromagnetic field propagation in media. The uniqueness of the solution and its behavior as t are proved. We also prove that the finite difference approximation converges in space and time. Numerical implementation with various experiments for different values of the input parameters is performed to validate the theoretical conclusions.

MSC 2010: 65M06; 35K55; 45K05

Award Identifier / Grant number: FR-21-2101

Funding statement: This research was supported by the Shota Rustaveli National Science Foundation of Georgia under the grant FR-21-2101.

References

[1] M. M. Aptsiauri, T. A. Dzhangveladze and Z. V. Kiguradze, Asymptotic behavior of the solution of a system of nonlinear integro-differential equations (in Russian), Differ. Uravn. 48 (2012), no. 1, 70–78; translation in Differ. Equ. 48 (2012), no. 1, 72–80. Search in Google Scholar

[2] Y. Bai and P. Zhang, On a class of Volterra nonlinear equations of parabolic type, Appl. Math. Comput. 216 (2010), no. 1, 236–240. 10.1016/j.amc.2010.01.044Search in Google Scholar

[3] T. A. Dzhangveladze, The first boundary value problem for a nonlinear equation of parabolic type (in Russian), Dokl. Akad. Nauk SSSR 269 (1983), no. 4, 839–842; translation in Soviet Phys. Dokl. 28 (1983), 323–324. Search in Google Scholar

[4] T. A. Dzhangveladze and Z. V. Kiguradze, Asymptotic behavior of a solution of a nonlinear system of the diffusion of a magnetic field into a substance (in Russian), Sibirsk. Mat. Zh. 47 (2006), no. 5, 1058–1070; translation in Siberian Math. J. 47 (2006), no. 5, 867–878. Search in Google Scholar

[5] D. G. Gordeziani, T. A. Dzhangveladze and T. K. Korshiya, Existence and uniqueness of the solution of a class of nonlinear parabolic problems (in Russian), Differ. Uravn. 19 (1983), no. 7, 1197–1207; translation in Differ. Equ. 19 (1983), 887–895. Search in Google Scholar

[6] F. Hecht, T. Jangveladze, Z. Kiguradze and O. Pironneau, Finite difference scheme for one system of nonlinear partial integro-differential equations, Appl. Math. Comput. 328 (2018), 287–300. 10.1016/j.amc.2018.01.050Search in Google Scholar

[7] T. Jangveladze, Convergence of a difference scheme for a nonlinear integro-differential equation, Proc. I. Vekua Inst. Appl. Math. 48 (1998), 38–43. Search in Google Scholar

[8] T. Jangveladze, Investigation and numerical solution of nonlinear partial differential and integro-differential models based on system of Maxwell equations, Mem. Differ. Equ. Math. Phys. 76 (2019), 1–118. Search in Google Scholar

[9] T. Jangveladze and Z. Kiguradze, Asymptotic properties of solution and difference scheme for one nonlinear integro-differential model, AMINSE 2017: Mathematics, Informatics, and Their Applications in Natural Sciences and Engineering, Springer Proc. Math. Stat. 246, Springer, Cham (2019), 117–133. 10.1007/978-3-030-10419-1_7Search in Google Scholar

[10] T. Jangveladze, Z. Kiguradze and B. Neta, Finite difference approximation of a nonlinear integro-differential system, Appl. Math. Comput. 215 (2009), no. 2, 615–628. 10.1016/j.amc.2009.05.061Search in Google Scholar

[11] T. Jangveladze, Z. Kiguradze and B. Neta, Large time asymptotic and numerical solution of a nonlinear diffusion model with memory, Comput. Math. Appl. 59 (2010), no. 1, 254–273. 10.1016/j.camwa.2009.07.052Search in Google Scholar

[12] T. Jangveladze, Z. Kiguradze and B. Neta, Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equations, Elsevier/Academic, Amsterdam, 2016. Search in Google Scholar

[13] T. A. Jangveladze and Z. V. Kiguradze, Asymptotics for large time of solutions to nonlinear system associated with the penetration of a magnetic field into a substance, Appl. Math. 55 (2010), no. 6, 471–493. 10.1007/s10492-010-0019-3Search in Google Scholar

[14] Z. Kiguradze, Finite difference scheme for a nonlinear integro-differential system, Proc. I. Vekua Inst. Appl. Math. 50(51) (2000/01), 65–72. Search in Google Scholar

[15] L. D. Landau and E. M. Lifšic, Electrodynamics of Continuous Media (in Russian), Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1957. Search in Google Scholar

[16] G. I. Laptev, Quasilinear parabolic equations that have a Volterra operator in the coefficients (in Russian), Mat. Sb. (N. S.) 136(178) (1988), no. 4, 530–545; translation in Math. USSR-Sb. 64 (1989), no. 2, 527–542. Search in Google Scholar

[17] Y. P. Lin and H.-M. Yin, Nonlinear parabolic equations with nonlinear functionals, J. Math. Anal. Appl. 168 (1992), no. 1, 28–41. 10.1016/0022-247X(92)90187-ISearch in Google Scholar

[18] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. Search in Google Scholar

[19] N. T. Long and P. N. D. Alain, Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance, Math. Methods Appl. Sci. 16 (1993), no. 4, 281–295. 10.1002/mma.1670160404Search in Google Scholar

[20] W. C. Rheinboldt, Methods for solving systems of nonlinear equations, Conf. Board Math. Sci. Reg. Conf. Ser. Appl. Math. 14, Society for Industrial and Applied Mathematics, Philadelphia, 1974. Search in Google Scholar

[21] A. A. Samarskiĭ, Theory of Difference Schemes (in Russian), Izdat. “Nauka”, Moscow, 1977. Search in Google Scholar

[22] G. D. Thornton, B. R. Anderson, M. A. Baugh, G. M. Robertson, J. Shapiro, R. C. Thyberg and B. Neta, Numerical solution of a nonlinear diffusion model with memory, Technical Report, Naval Postgraduate School, Monterey, 2018. Search in Google Scholar

[23] M. I. Višik, Solubility of boundary-value problems for quasi-linear parabolic equations of higher orders, Mat. Sb. (N. S.) 59 (101) (1962), 289–325. Search in Google Scholar

Received: 2022-07-23
Accepted: 2022-09-26
Published Online: 2023-04-12
Published in Print: 2023-08-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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