Numerical solution of fractional advection-diffusion equation with generalized Caputo derivative using shifted ultraspherical collocation method

Authors

  • K. Issa
    Department of Mathematics and Statistics, Kwara State University, Malete, Nigeria
  • A. T. AbdulKareem
    Department of Mathematics and Statistics, Kwara State University, Malete, Nigeria
  • A D. Adeshola
    Department of Mathematics and Statistics, Kwara State University, Malete, Nigeria
  • R. A. Bello
    Department of Mathematics and Statistics, Kwara State University, Malete, Nigeria

Keywords:

Ultraspherical collocation method, Generalized Caputo Fractional Derivative (GCFD), Shifted ultraspherical polynomial, Fractional advection-diffusion equation, Finite difference scheme

Abstract

In this article, we develop an approximate technique for solving a fractional advection-diffusion equation with a generalized Caputo derivative via a finite difference scheme. This technique employed the properties of shifted ultraspherical polynomials, which combine the finite difference scheme for the temporal discretization derivative and express the approximate solution in terms of shifted ultraspherical polynomials. The convergence and the stability of the proposed technique are proved. Numerical examples are considered, and the obtained results are compared with the analytical results and those obtained in the literature to establish the accuracy of the technique. The proposed techniques, based on the generalized Caputo derivative and the properties of shifted ultraspherical polynomials, produce robust results and encompass those obtained using several other families of orthogonal polynomials as demonstrated in the results of the examples considered.

Dimensions

[1] J. T. Machado, V. Kiryakova & F. Mainardi, “Recent history of fractional calculus”, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 1140. https://doi.org/10.1016/j.cnsns.2010.05.027.

[2] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, CA, 1998, pp. 123–135. https://lib.ugent.be/catalog/rug01:002178612.

[3] K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, Springer, Berlin, Germany, 2010. https://doi.org/10.1007/978-3-642-14574-2.

[4] M. Sadki, S. Harroudi & K. Allali, “Fractional-order SIR epidemic model with treatment cure rate”, Part. Differ. Equ. Appl. Math. 8 (2023) 100593. https://doi.org/10.1016/j.padiff.2023.100593.

[5] Y. Zhang, L. Zhang & X. Tian, “Conservation laws for systems of non-standard Birkhoffians with fractional derivatives”, Commun. Nonlinear Sci. Numer. Simul. 130 (2024) 107722. https://doi.org/10.1016/j.cnsns.2023.107722.

[6] N. Ramroodi, H. A. Tehrani & M. H. Noori Skandari, “Numerical behavior of the variable-order fractional Van der Pol oscillator”, J. Comput Sci. 74 (2023) 102174. https://doi.org/10.1016/j.jocs.2023.102174.

[7] H. Wang & N. Yamamoto, “Using a partial differential equation with Google mobility data to predict COVID-19 in Arizona”, Mathematical Biosciences and Engineering 17 (2020) 4891. https://doi.org/10.3934/mbe.2020266.

[8] H. Ahmad, A. Akgül, T. A. Khan, P. S. Stanimirovi & Y. Chu, “New perspective on the conventional solutions of the nonlinear time-fractional partial differential equations”, Complexity 2020 (2020) 8829017. https://doi.org/10.1155/2020/8829017.

[9] A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli, T. J. R. Hughes, A. Patton, A. Reali, T. E. Yankeelov & A. Veneziani, “Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-diseased (SEIRD) model with heterogeneous diffusion”, Appl Math. Lett. 111 (2021) 106617. https://doi.org/10.1016/j.aml.2020.106617.

[10] B. A. Carreras, V. E. Lynch & G. M. Zaslavsky, “Anomalous diffusion and exit time distribution of particle travers in plasma turbulence models”, Phys. Plasma 8 (2001) 5096. https://doi.org/10.1063/1.1416180.

[11] D. Benson, S. W. Wheatcraft & M. M. Meerschaert, “Application of a fractional advection-dispersion equation”, Water Resour. Res. 36 (2000) 1403. https://doi.org/10.1029/2000WR900031.

[12] K. Issa, B. M. Yisa & J. Biazar, “Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials”, Computational Methods for Differential Equations 10 (2022) 431. https://tinyurl.com/54mrdm2y.

[13] N. H. Sweilam, A. M. Nagy & A. A. El-Sayed, “Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation”, Chaos, Solitons & Fractals 73 (2015) 141. https://doi.org/10.1016/j.chaos.2015.01.010.

[14] Y. E. Aghdam, H. Safdari & M. Javidi, “Numerical approach of the space fractional order diffusion equation based on the third kind of Chebyshev polynomials”, Combinatorics, Crytography, Computer Sci. & Computing, Conference paper 2019. https://tinyurl.com/24bc5832.

[15] N. H. Sweilam, A. M. Nagy & A. A. El-Sayed, “Numerical approach for solving space fractional order diffusion equation using shifted Chebyshev polynomials of the fourth kind”, Turkish Journal of Mathematics 40 (2016) 1283. https://doi.org/10.3906/mat-1503-20.

[16] K. Issa, A. S. Olorunnisola, T. O. Aliu & A. D. Adeshola, “Approximate solution of space fractional order diffusion equations by Gegenbauer collocation and compact finite difference scheme”, J. Nig. Soc. Phys. Sci. 5 (2023) 1368. https://doi.org/10.46481/jnsps.2023.1368.

[17] Y. E. Aghdam, H. Safdari, Y. Azari, H. Jafari & D. Baleanu, “Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme”, Discrete & Continuous Dynamical Systems-S 14 (2021) 20. https://doi.org/10.3934/dcdss.2020402.

[18] S. Kumar, K. Kumar, R. K. Pandey & Y. Xu, “Legendre collocation method for new generalized fractional advection-diffusion equation”, Int. J. Comput. Math 101 (2024) 1. https://doi.org/10.1080/00207160.2024.2305640.

[19] A. Jannelli, M. Ruggieri & M. P. Speciale, “Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term”, Appl. Numer. Math. 155 (2020) 93. https://doi.org/10.1016/j.apnum.2020.01.016.

[20] N. Attia, A. Akgül, D. Seba & A. Nour, “On solutions of time-fractional advection-diffusion equation”, Numer. Methods Part. Differ. Equ. 39 (2023) 4489. https://doi.org/10.1002/num.22621.

[21] A. M. Nagy & K. Issa, “An accurate numerical technique for solving fractional advection-diffusion equation with generalized Caputo derivative”, ZAMP 75 (2024) 164. https://doi.org/10.1007/s00033-024-02309-5.

[22] H. Tajadodi, “A numerical approach of fractional advection-diffusion equation with Atangana-Baleanu derivative”, Chaos Solitons Fractals 130 (2020) 109527. https://doi.org/10.1016/j.chaos.2019.109527.

[23] R. Almeida, A. B. Malinowska & T. Odzijewicz, “Fractional differential equations with dependence on the Caputo-Katugampola derivative”, J. Comput. Nonlinear Dyn. 11 (2016) 061017. https://doi.org/10.1115/1.4034432.

[24] N. H. Sweilam, A. M. Nagy & T. M. Al-Ajami, “Numerical solutions of fractional optimal control with Caputo-Katugampola derivative”, Adv. Differ. Equ. 2021 (2021) 425. https://doi.org/10.1186/s13662-021-03580-w.

[25] G. Szego, Orthogonal polynomials, 4th Ed., AMS Colloq. Publ., 1975. http://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf.

[26] V. J. Ervin & J. P. Roop, “Variational solution of fractional advection dispersion equations on bounded domains in Rd”, Numer. Meth. Part D 23 (2007) 256. http://dx.doi.org/10.1002/num.20169.

Published

2026-05-02

How to Cite

Numerical solution of fractional advection-diffusion equation with generalized Caputo derivative using shifted ultraspherical collocation method. (2026). Journal of the Nigerian Society of Physical Sciences, 8(2), 2986. https://doi.org/10.46481/jnsps.2026.2986

Issue

Volume 8, Issue 2, May 2026

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2026 K. Issa, A. T. AbdulKareem, A D. Adeshola, R. A. Bello (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Numerical solution of fractional advection-diffusion equation with generalized Caputo derivative using shifted ultraspherical collocation method. (2026). Journal of the Nigerian Society of Physical Sciences, 8(2), 2986. https://doi.org/10.46481/jnsps.2026.2986

Similar Articles

31-40 of 250

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)