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Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo–Fabrizio fractional derivative

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  • Wei, Leilei
  • Li, Wenbo

Abstract

In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order α(t)∈(0,1), we derive the stability and L2 convergence of proposed scheme and prove that the method is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.

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  • Wei, Leilei & Li, Wenbo, 2021. "Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo–Fabrizio fractional derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 280-290.
    • Handle: RePEc:eee:matcom:v:188:y:2021:i:c:p:280-290
    DOI: 10.1016/j.matcom.2021.04.001
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    References listed on IDEAS

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    1. Haq, Sirajul & Ghafoor, Abdul & Hussain, Manzoor, 2019. "Numerical solutions of variable order time fractional (1+1)- and (1+2)-dimensional advection dispersion and diffusion models," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 107-121.
    2. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
    3. Sun, HongGuang & Chen, Wen & Chen, YangQuan, 2009. "Variable-order fractional differential operators in anomalous diffusion modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4586-4592.
    4. Emile Franc Doungmo Goufo & Sunil Kumar, 2017. "Shallow Water Wave Models with and without Singular Kernel: Existence, Uniqueness, and Similarities," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-9, February.
    5. Li, Changpin & Wang, Zhen, 2021. "Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 838-857.
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    1. Fouladi, Somayeh & Dahaghin, Mohammad Shafi, 2022. "Numerical investigation of the variable-order fractional Sobolev equation with non-singular Mittag–Leffler kernel by finite difference and local discontinuous Galerkin methods," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    2. Wei, Leilei & Wang, Huanhuan, 2023. "Local discontinuous Galerkin method for multi-term variable-order time fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 685-698.

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