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A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method

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  • Khader, M.M.
  • Saad, K.M.

Abstract

In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. This equation presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce the proposed problem to a system of ODEs, which is solved by using finite difference method (FDM). Some theorems about the convergence analysis are stated. A numerical simulation and a comparison with the previous work are presented. We can apply the proposed method to solve other problems in engineering and physics.

Suggested Citation

  • Khader, M.M. & Saad, K.M., 2018. "A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method," Chaos, Solitons & Fractals, Elsevier, vol. 110(C), pages 169-177.
    • Handle: RePEc:eee:chsofr:v:110:y:2018:i:c:p:169-177
    DOI: 10.1016/j.chaos.2018.03.018
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    References listed on IDEAS

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    1. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
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    3. Gómez-Aguilar, J.F., 2017. "Irving–Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 179-186.
    4. Mohammad Maleki & Ishak Hashim & Majid Tavassoli Kajani & Saeid Abbasbandy, 2012. "An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, December.
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    Cited by:

    1. Heydari, M.H. & Razzaghi, M. & Avazzadeh, Z., 2021. "Orthonormal shifted discrete Chebyshev polynomials: Application for a fractal-fractional version of the coupled Schrödinger-Boussinesq system," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    2. Saad, Khaled M., 2021. "Fractal-fractional Brusselator chemical reaction," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    3. Singh, C.S. & Singh, Harendra & Singh, Somveer & Kumar, Devendra, 2019. "An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 1440-1448.
    4. H. Çerdik Yaslan, 2021. "Numerical solution of the nonlinear conformable space–time fractional partial differential equations," Indian Journal of Pure and Applied Mathematics, Springer, vol. 52(2), pages 407-419, June.
    5. Khader, M.M. & Inc, Mustafa, 2021. "Numerical technique based on the interpolation with Lagrange polynomials to analyze the fractional variable-order mathematical model of the hepatitis C with different types of virus genome," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    6. Maryam Al Owidh & Basma Souayeh & Imran Qasim Memon & Kashif Ali Abro & Huda Alfannakh, 2022. "Heat Transfer and Fluid Circulation of Thermoelectric Fluid through the Fractional Approach Based on Local Kernel," Energies, MDPI, vol. 15(22), pages 1-12, November.
    7. Cayama, Jorge & Cuesta, Carlota M. & de la Hoz, Francisco, 2021. "A pseudospectral method for the one-dimensional fractional Laplacian on R," Applied Mathematics and Computation, Elsevier, vol. 389(C).
    8. Agarwal, P. & Deni̇z, S. & Jain, S. & Alderremy, A.A. & Aly, Shaban, 2020. "A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 542(C).
    9. Khater, Mostafa M.A. & Mohamed, Mohamed S. & Attia, Raghda A.M., 2021. "On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    10. Hari Mohan Srivastava & Khaled M. Saad, 2020. "A Comparative Study of the Fractional-Order Clock Chemical Model," Mathematics, MDPI, vol. 8(9), pages 1-14, August.

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