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Finite element approximations of a nonlinear diffusion model with memory

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Abstract

The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.

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References

  1. Landau, L., Lifschitz, E.: Electrodynamics of Continuous Media. Course of Theoretical Physics, Vol. 8. Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass, 1960, Russian original: Gosudarstv. Izdat. Tehn-Teor. Lit., Moscow (1957, translated from the Russian)

  2. Gordeziani, D., Dzhangveladze, T., Korshia, T.: Existence and uniqueness of the solution of a class of nonlinear parabolic problems. Differ. Uravn. 19, 1197–1207 (1983, Russian). Diff. Equ. 19, 887–895 (1983, English translation)

    MATH  Google Scholar 

  3. Dzhangveladze, T.A.: First boundary value problem for a nonlinear equation of parabolic type. Dokl. Akad. Nauk SSSR 269, 839–842 (1983, Russian). Sov. Phys. Dokl. 28, 323–324 (1983, English translation)

    MATH  Google Scholar 

  4. Dzhangveladze, T.: An Investigation of the First Boundary Value Problem for Some Nonlinear Integro-differential Equations of Parabolic Type. Tbilisi State University, (Russian) (1983)

  5. Dzhangveladze, T.A.: A nonlinear integro-differential equation of parabolic type. Differ. Uravn. 21, 41–46 (1985, Russian). Diff. Equ. 21, 32–36 (1985, English translation)

    MathSciNet  MATH  Google Scholar 

  6. Laptev, G.: Quasilinear parabolic equations which contain in coefficients Volterra’s operator. Math. Sbornik 136, 530–545 (1988, Russian). Sbornik Math. 64, 527–542 (1989, English translation)

    MathSciNet  Google Scholar 

  7. Lin, Y., Yin, H.M.: Nonlinear parabolic equations with nonlinear functionals. J. Math. Anal. Appl. 168, 28–41 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jangveladze, T.: On one class of nonlinear integro-differential equations. Semin. I. Vekua Inst. Appl. Math. 23, 51–87 (1997)

    MathSciNet  Google Scholar 

  9. Vishik, M.: Solvability of boundary-value problems for quasi-linear parabolic equations of higher orders (Russian). Math. Sb. (N.S.) 59(101), suppl. 289–325 (1962)

    Google Scholar 

  10. Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires. Dunod, Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  11. Kiguradze, Z.: The asymptotic behavior of the solutions of one nonlinear integro-differential model. Semin. I. Vekua Inst. Appl. Math. Reports 30, 21–32 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Kiguradze, Z.V.: Asymptotic behavior and numerical solution of the system of nonlinear integro-differential equations. Rep. Enl. Sess. Sem. I. Vekua Inst. Appl. Math. 19, 58–61 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Jangveladze, T.A., Kiguradze, Z.V.: Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance. Sib. Mat. Z. 47, 1058–1070 (2006, Russian). Sib. Math. J. 47, 867–878 (2006, English translation)

    Article  Google Scholar 

  14. Jangveladze, T., Kiguradze, Z., Neta, B.: Large time behavior of solutions to a nonlinear integro-differential system. J. Math. Anal. Appl. 351, 382–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jangveladze, T., Kiguradze, Z., Neta, B.: Large time asymptotic and numerical solution of a nonlinear diffusion model with memory. Comput. Math. Appl. 59, 254–273 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jangveladze, T., Kiguradze, Z., Neta, B.: Large time behavior of solutions and finite difference scheme to a nonlinear integro-differential equation. Comput. Math. Appl. 57, 799–811 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jangveladze, T.: Convergence of a difference scheme for a nonlinear integro-differential equation. Proc. I. Vekua Inst. Appl. Math. 48, 38–43 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Kiguradze, Z.: Finite difference scheme for a nonlinear integro-differential system. Proc. I. Vekua Inst. Appl. Math 50–51, 65–72 (2000–2001)

    MathSciNet  Google Scholar 

  19. Jangveladze, T., Kiguradze, Z., Neta, B.: Finite difference approximation of a nonlinear integro-differential system. Appl. Math. Comput. 215, 615–628 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Neta, B.: Numerical solution of a nonlinear integro-differential equation. J. Math. Anal. Appl. 89, 598–611 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Neta, B., Igwe, J.O.: Finite differences versus finite elements for solving nonlinear integro-differential equations. J. Math. Anal. Appl. 112, 607–618 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jangveladze, T., Kiguradze, Z., Neta, B.: Galerkin finite element method for one nonlinear integro-differential model. Appl. Math. Comput. 217, 6883–6892 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method, 2nd edn. Wellesley-Cambridge Press, Wellesley, MA (2008)

    Google Scholar 

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Authors and Affiliations

  1. Ilia Vekua Institute of Applied Mathematics, Ivane Javakhishvili Tbilisi State University, 2 University Street, 0186, Tbilisi, Georgia

    Temur Jangveladze & Zurab Kiguradze

  2. Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 3 Chavchavdze Ave., 0179, Tbilisi, Georgia

    Temur Jangveladze & Zurab Kiguradze

  3. Caucasus University, 77 Kostava Ave., 0175, Tbilisi, Georgia

    Temur Jangveladze

  4. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, 93943, USA

    Beny Neta

  5. Department of Mathematics, The Technion–Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

Authors
  1. Temur Jangveladze
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  2. Zurab Kiguradze
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  3. Beny Neta
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  4. Simeon Reich
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Corresponding author

Correspondence to Beny Neta.

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Jangveladze, T., Kiguradze, Z., Neta, B. et al. Finite element approximations of a nonlinear diffusion model with memory. Numer Algor 64, 127–155 (2013). https://doi.org/10.1007/s11075-012-9658-7

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  • DOI: https://doi.org/10.1007/s11075-012-9658-7

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