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Numerical solutions of variable order time fractional (1+1)- and (1+2)-dimensional advection dispersion and diffusion models

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  • Haq, Sirajul
  • Ghafoor, Abdul
  • Hussain, Manzoor

Abstract

A numerical scheme based on Haar wavelets coupled with finite differences is suggested to study variable order time fractional partial differential equations (TFPDEs). The technique is tested on (1 + 1)-dimensional advection dispersion and (1 + 2)-dimensional advection diffusion equations. In the proposed scheme, time fractional derivative is firstly approximated by quadrature formula, and then finite differences are combined with one and two dimensional Haar wavelets. With the help of suggested method the TFPDEs convert to a system of algebraic equations which is easily solvable. Also convergence of the proposed scheme has been discussed which is an important part of the present work. For validation, the obtained results are matched with earlier work and exact solutions. Computations illustrate that the proposed scheme has better outcomes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:360:y:2019:i:c:p:107-121
    DOI: 10.1016/j.amc.2019.04.085
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    References listed on IDEAS

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    1. Arbabi, Somayeh & Nazari, Akbar & Darvishi, Mohammad Taghi, 2017. "A two-dimensional Haar wavelets method for solving systems of PDEs," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 33-46.
    2. Tianzeng Li & Yu Wang & Yong Yang, 2014. "Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-14, April.
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    Cited by:

    1. Abdul Ghafoor & Sirajul Haq & Manzoor Hussain & Poom Kumam & Muhammad Asif Jan, 2019. "Approximate Solutions of Time Fractional Diffusion Wave Models," Mathematics, MDPI, vol. 7(10), pages 1-15, October.
    2. 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.
    3. Sadri, Khadijeh & Aminikhah, Hossein, 2021. "An efficient numerical method for solving a class of variable-order fractional mobile-immobile advection-dispersion equations and its convergence analysis," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    4. 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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