A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation
Abstract
The purpose of the article is to analyze an accurate numerical technique to solve a space-time fractional-order Fisher equation in the Caputo sense. For this purpose, the spectral collocation technique is used, which is based on the Vieta–Lucas approximation. By using the properties of Vieta–Lucas polynomials, this technique reduces the nonlinear equations into a system of ordinary differential equations (ODEs). The non-standard finite difference (NSFD) method converts this system of ODEs into algebraic equations which have been solved numerically. Moreover, the error estimate is investigated for the proposed method. To show the accuracy and efficiency of the technique, the obtained numerical results are compared with the analytical results and existing results of the particular forms of the considered fractional order models through error analysis. The important feature of this article is the exhibition of variations of the field variable for various values of spatial and temporal fractional order parameters for different particular cases.
Keyword : spectral collocation method, space-time fractional Fisher equation, Vieta–Lucas polynomials, NSFD method
How to Cite
Kashif, M. (2025). A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation. Mathematical Modelling and Analysis, 30(1), 1–16. https://doi.org/10.3846/mma.2025.19839
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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P. Singh and D. Sharma. Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE. Nonlinear Eng., 9(1):60–71, 2020. https://doi.org/10.1515/nleng-2018-0136
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Y.E. Aghdam, H. Mesgarani, Z. Asadi and V.T. Nguyen. Investigation and analysis of the numerical approach to solve the multi-term timefractional advection-diffusion model. AIMS Math., 8(12):29474–29489, 2023. https://doi.org/10.3934/math.20231509
H. Ahmad, A.R. Seadawy, T.A. Khan and P. Thounthong. Analytic approximate solutions for some nonlinear parabolic dynamical wave equations. J. Taibah Univ. Sci., 14(1):346–358, 2020. https://doi.org/10.1080/16583655.2020.1741943
K. Al-Khaled. Numerical study of Fisher’s reaction–diffusion equation by the Sinc collocation method. J. Comput. Appl. Math., 137(2):245–255, 2001. https://doi.org/10.1016/S0377-0427(01)00356-9
M. Al-Qurashi, Z. Korpinar, D. Baleanu and M. Inc. A new iterative algorithm on the time-fractional Fisher equation: Residual power series method. Adv. Mech. Eng., 9(9):1687814017716009, 2017. https://doi.org/10.1177/1687814017716009
R.L. Bagley and P.J. Torvik. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27(3):201–210, 1983. https://doi.org/10.1122/1.549724
V.K. Baranwal, R.K. Pandey, M.P. Tripathi and O.P. Singh. An analytic algorithm for time fractional nonlinear reaction–diffusion equation based on a new iterative method. Commun. Nonlinear Sci. Numer. Simul., 17(10):3906–3921, 2012. https://doi.org/10.1016/j.cnsns.2012.02.015
M.H. Cherif, K. Belghaba and D. Ziane. Homotopy perturbation method for solving the fractional Fisher’s equation. Int. J. Anal. Appl., 10(1):9–16, 2016.
H. Dehestani, Y. Ordokhani and M. Razzaghi. A spectral approach for timefractional diffusion and subdiffusion equations in a large interval. Math. Model. Anal., 27(1):19–40, 2022. https://doi.org/10.3846/mma.2022.13579
A.Y. Esmaeelzade, H. Mesgarani and Z. Asadi. Estimate of the fractional advection-diffusion equation with a time-fractional term based on the shifted Legendre polynomials. J. Math. Model., 11(4):731–744, 2023. https://doi.org/10.22124/jmm.2023.24479.2191
R.A. Fisher. The wave of advance of advantageous genes. Ann. eugen., 7(4):355– 369, 1937. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto. Fractional calculus and continuous-time finance III: the diffusion limit. In Mathematical Finance: Workshop of the Mathematical Finance Research Project, pp. 171–180, Germany, 2001. Springer.
H. Jafari, Y.E. Aghdam, B. Farnam, V.T. Nguyen and M.T. Masetshaba. A convergence analysis of the mobile-immobile advection-dispersion model of temporal fractional order arising in watershed catchments and rivers. Fractals, 31(4):2340068, 2023. https://doi.org/10.1142/S0218348X23400686
M. Kashif, K.D. Dwivedi and T. Som. Numerical solution of coupled type fractional order Burgers’ equation using finite difference and Fibonacci collocation method. Chin. J. Phys., 77:2314–2323, 2022. https://doi.org/10.1016/j.cjph.2021.10.044
M.M. Khader and M. Adel. Numerical and theoretical treatment based on the compact finite difference and spectral collocation algorithms of the space fractional-order Fisher’s equation. Int. J. Mod. Phys. C, 31(09):2050122, 2020. https://doi.org/10.1142/S0129183120501223
M.M. Khader, J.E. Macías-Díaz, K.M. Saad and W.M. Hamanah. Vieta–Lucas polynomials for the Brusselator system with the Rabotnov fractionalexponential kernel fractional derivative. Symmetry, 15(9):1619, 2023. https://doi.org/10.3390/sym15091619
M.M. Khader and K.M. Saad. A numerical approach for solving the problem of biological invasion (fractional Fisher equation) using Chebyshev spectral collocation method. Chaos Solit. Fractals, 110:169–177, 2018. https://doi.org/10.1016/j.chaos.2018.03.018
A. Majeed, M. Kamran, M. Abbas and J. Singh. An efficient numerical technique for solving time-fractional generalized Fisher’s equation. Front. Phys., 8:293, 2020. https://doi.org/10.3389/fphy.2020.00293
H. Marasi and M. Derakhshan. A composite collocation method based on the fractional Chelyshkov wavelets for distributed-order fractional mobileimmobile advection-dispersion equation. Math. Model. Anal., 27(4):590–609, 2022. https://doi.org/10.3846/mma.2022.15311
H. Mesgarani, Y.E. Aghdam, M. Khoshkhahtinat and B. Farnam. Analysis of the numerical scheme of the one-dimensional fractional Rayleigh–Stokes model arising in a heated generalized problem. AIP Advances, 13(8), 2023. https://doi.org/10.1063/5.0156586
H. Mesgarani, A.Y. Esmaeelzade and M. Vafapisheh. A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis. AIP Advances, 13(5), 2023. https://doi.org/10.1063/5.0143690
R.E. Mickens. Applications of nonstandard finite difference schemes. World Scientific, 2000.
S.T. Mohyud-Din and M.A. Noor. Modified variational iteration method for solving Fisher’s equations. J. Appl. Math. Comput., 31:295–308, 2009. https://doi.org/10.1007/s12190-008-0212-7
K.M. Saad, M.M. Khader, J.F. Gómez-Aguilar and D. Baleanu. Numerical solutions of the fractional Fisher’s type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods. Chaos, 29(2), 2019. https://doi.org/10.1063/1.508677
A. Şahin, İ. Dǎg and B. Saka. AB-spline algorithm for the numerical solution of Fisher’s equation. Kybernetes, 37(2):326–342, 2008. https://doi.org/10.1108/03684920810851212
R. Schumer, D.A. Benson, M.M. Meerschaert and S.W. Wheatcraft. Eulerian derivation of the fractional advection–dispersion equation. J. Contam. Hydrol., 48(1-2):69–88, 2001. https://doi.org/10.1016/S0169-7722(00)00170-4
A. Secer and M. Cinar. A Jacobi wavelet collocation method for fractional Fisher’s equation in time. Therm. Sci., 24(Suppl. 1):119–129, 2020. https://doi.org/10.2298/TSCI20S1119S
P. Singh and D. Sharma. Comparative study of homotopy perturbation transformation with homotopy perturbation Elzaki transform method for solving nonlinear fractional PDE. Nonlinear Eng., 9(1):60–71, 2020. https://doi.org/10.1515/nleng-2018-0136
A.K. Verma, M.K. Rawani and C. Cattani. A numerical scheme for a class of generalized Burgers’ equation based on Haar wavelet nonstandard finite difference method. Appl. Numer. Math., 168:41–54, 2021. https://doi.org/10.1016/j.apnum.2021.05.019
A.-M. Wazwaz and A. Gorguis. An analytic study of Fisher’s equation by using Adomian decomposition method. Appl. Math. Comput., 154(3):609–620, 2004. https://doi.org/10.1016/S0096-3003(03)00738-0