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Numerical simulation for the space-fractional diffusion equations

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  • Kheybari, Samad
  • Darvishi, Mohammad Taghi
  • Hashemi, Mir Sajjad

Abstract

This paper presents, a novel semi-analytical algorithm, based on the Chebyshev collocation method, for the solution of space-fractional diffusion equations. The original fractional equation is transformed into a system of ordinary differential equations (ODEs) by the Chebyshev collocation method. A new semi-analytical method is then used to approximate the solution of the resulting system. To emphasize the reliability of the new scheme, a convergence analysis is presented. It is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased. By presenting some numerical examples, we show that the proposed method is a powerful and reliable algorithm for solving space-fractional diffusion equations and can be extended to solve another space-fractional partial differential equations. Comparison with other methods in the literature, demonstrates that the proposed method is both efficient and accurate.

Suggested Citation

  • Kheybari, Samad & Darvishi, Mohammad Taghi & Hashemi, Mir Sajjad, 2019. "Numerical simulation for the space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 57-69.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:57-69
    DOI: 10.1016/j.amc.2018.11.041
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    References listed on IDEAS

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    1. Kheybari, S. & Darvishi, M.T., 2018. "An efficient technique to find semi-analytical solutions for higher order multi-point boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 76-93.
    2. Lin, Ji & Reutskiy, S.Y. & Lu, Jun, 2018. "A novel meshless method for fully nonlinear advection–diffusion-reaction problems to model transfer in anisotropic media," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 459-476.
    3. Sweilam, N.H. & Nagy, A.M. & El-Sayed, Adel A., 2015. "Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 141-147.
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    Cited by:

    1. Kheybari, Samad, 2021. "Numerical algorithm to Caputo type time–space fractional partial differential equations with variable coefficients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 66-85.
    2. Hashemi, M.S., 2021. "A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Darvishi, M.T. & Najafi, Mohammad & Wazwaz, Abdul-Majid, 2021. "Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    4. Chaudhary, Manish & Kumar, Rohit & Singh, Mritunjay Kumar, 2020. "Fractional convection-dispersion equation with conformable derivative approach," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).

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