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Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time Derivative

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Abstract

A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with Caputo fractional time derivative is considered. The coefficients of the elliptic part of the equation depend on the derivatives of the solution and satisfy the conditions providing strong monotonicity and Lipschitz-continuity of the corresponding operator. The equation is approximated by two finite-difference schemes: implicit and fractional step scheme. The stability of these finite difference schemes is proved and accuracy estimates are obtained under the condition of sufficient smoothness of the input data and the solution of the differential problem. A number of iterative methods for implementing the constructed nonlinear mesh problems are analyzed. The convergence and convergence rate of the iterative methods are substantiated. The results of numerical experiments confirming the theoretical conclusions are presented.

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Notes

  1. Further we will not distinguish between the mesh functions and the vectors of their nodal values, as well as the mesh operators and the corresponding matrices.

  2. For a parabolic problem with integer derivatives, a locally one-dimensional scheme of this kind was first proposed in [25].

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Funding

This work was supported by the Russian Foundation for Basic Researches, project no. 19-01-00431.

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Authors and Affiliations

  1. Institute of Computational Mathematics and Information Technologies, Kazan (Volga Region) Federal University, 420008, Kazan, Tatarstan, Russia

    A. V. Lapin & K. O. Levinskaya

  2. I. M. Sechenov First Moscow State Medical University, 119992, Moscow, Russia

    A. V. Lapin

Authors
  1. A. V. Lapin
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  2. K. O. Levinskaya
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Correspondence to A. V. Lapin or K. O. Levinskaya.

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Submitted by A. M. Elizarov)

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Lapin, A.V., Levinskaya, K.O. Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time Derivative. Lobachevskii J Math 41, 2673–2686 (2020). https://doi.org/10.1134/S1995080220120215

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  • DOI: https://doi.org/10.1134/S1995080220120215

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