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A computational method for solving variable-order fractional nonlinear diffusion-wave equation

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  • Heydari, Mohammad Hossein
  • Avazzadeh, Zakieh
  • Yang, Yin

Abstract

In this paper, we generalize a one-dimensional fractional diffusion-wave equation to a one-dimensional variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TFND-WE) where the variable-order fractional derivatives are considered in the Caputo type. To solve this introduced equation, an easy-to-follow method is proposed which is based on the Chebyshev cardinal functions coupling with the tau and collocation methods. To carry out the method, an operational matrix of variable-order fractional derivative (OMV-OFD) is derived for the Chebyshev cardinal functions to be employed for expanding the unknown function. The proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling. The experimental results confirm the applicability and effectiveness of the method.

Suggested Citation

  • Heydari, Mohammad Hossein & Avazzadeh, Zakieh & Yang, Yin, 2019. "A computational method for solving variable-order fractional nonlinear diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 235-248.
    • Handle: RePEc:eee:apmaco:v:352:y:2019:i:c:p:235-248
    DOI: 10.1016/j.amc.2019.01.075
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    References listed on IDEAS

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    1. Heydari, Mohammad Hossein & Avazzadeh, Zakieh & Haromi, Malih Farzi, 2019. "A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 215-228.
    2. Heydari, Mohammad Hossein & Avazzadeh, Zakieh, 2018. "Legendre wavelets optimization method for variable-order fractional Poisson equation," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 180-190.
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    Cited by:

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    2. Heydari, M.H., 2020. "Chebyshev cardinal functions for a new class of nonlinear optimal control problems generated by Atangana–Baleanu–Caputo variable-order fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    3. Ahmed, Hoda F. & Hashem, W.A., 2023. "A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 388-408.
    4. Heydari, M.H. & Atangana, A., 2019. "A cardinal approach for nonlinear variable-order time fractional Schrödinger equation defined by Atangana–Baleanu–Caputo derivative," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 339-348.
    5. Hosseininia, M. & Heydari, M.H., 2019. "Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 389-399.
    6. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    7. Peng, Xiao & Wang, Yijing & Zuo, Zhiqiang, 2022. "Co-design of state-dependent switching law and control scheme for variable-order fractional nonlinear switched systems," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    8. Heydari, M. H. & Atangana, A., 2020. "An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).

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