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A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation

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  • Heydari, Mohammad Hossein
  • Avazzadeh, Zakieh
  • Haromi, Malih Farzi

Abstract

We firstly generalize a multi-term time fractional diffusion-wave equation to the multi-term variable-order time fractional diffusion-wave equation (M-V-TFD-E) by the concept of variable-order fractional derivatives. Then we implement the Chebyshev wavelets (CWs) through the operational matrix method to approximate its solution in the unit square. In fact, we apply the operational matrix of variable-order fractional derivative (OMV-FD) of the CWs to derive the unknown solution. We proceed with coupling the collocation and tau methods to reduce M-V-TFD-E to a system of algebraic equations. The important privilege of method is handling different kinds of conditions, i.e., initial-boundary conditions and Dirichlet boundary conditions, by implementing the same techniques. The convergence and error estimation of the CWs expansion in two dimensions are theoretically investigated. We also examine the applicability and computational efficiency of the new scheme through the numerical experiments.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:341:y:2019:i:c:p:215-228
    DOI: 10.1016/j.amc.2018.08.034
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    References listed on IDEAS

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    1. M. H. Heydari & M. R. Hooshmandasl & F. M. Maalek Ghaini & F. Mohammadi, 2012. "Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-19, February.
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    4. 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.
    5. Agarwal, P. & El-Sayed, A.A., 2018. "Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 500(C), pages 40-49.
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    2. Hosseininia, M. & Heydari, M.H., 2019. "Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 400-407.
    3. Sadri, Khadijeh & Amilo, David & Hinçal, Evren, 2025. "A combination of classical and shifted Jacobi polynomials for two-dimensional time-fractional diffusion-wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 198(C).
    4. 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).
    5. 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.
    6. 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.
    7. 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.
    8. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    9. Dehestani, H. & Ordokhani, Y. & Razzaghi, M., 2020. "Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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