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Approximate Solutions of Time Fractional Diffusion Wave Models

Author

Listed:
  • Abdul Ghafoor

    (Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan)

  • Sirajul Haq

    (Faculty of Engineering Sciences, GIK Institute, Topi 23640, KP, Pakistan)

  • Manzoor Hussain

    (Faculty of Engineering Sciences, GIK Institute, Topi 23640, KP, Pakistan)

  • Poom Kumam

    (Theoretical and Computational Science (TaCS) Center Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
    KMUTT-Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan)

  • Muhammad Asif Jan

    (Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan)

Abstract

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:923-:d:273285
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    References listed on IDEAS

    as
    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. Fengying Zhou & Xiaoyong Xu, 2017. "Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method," Advances in Mathematical Physics, Hindawi, vol. 2017, pages 1-17, September.
    3. 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.
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    Cited by:

    1. Lin Chen & Wenzhi Xu & Zhuojia Fu, 2022. "A Novel Spatial–Temporal Radial Trefftz Collocation Method for 3D Transient Wave Propagation Analysis with Specified Sound Source Excitation," Mathematics, MDPI, vol. 10(6), pages 1-15, March.
    2. Jiong Weng & Xiaojing Liu & Youhe Zhou & Jizeng Wang, 2022. "An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations," Mathematics, MDPI, vol. 10(21), pages 1-14, October.

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