Numerical solution of space fractional diffusion equation using shifted Gegenbauer polynomials

Document Type : Research Paper


1 Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria.

2 Department of Mathematics, University of Ilorin, Ilorin, Nigeria.

3 Department of Mathematical Sciences, University of Guilan, Rasht, Iran.


This paper is concerned with numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package. 


  • [1]          A. E. Abouelregal, S. Yao, and H. Ahmad, Analysis of a functionally graded thermopiezoelectric finite rod excited by a moving heat source, Results in Physics, 19 (2020), 103389.
  • [2]          W. M. Abd-Elhameed1 and Y. H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comp. Appl. Math., 37 (2018), 2897–2921.
  • [3]          Y. E. Aghdam, H. Safdari, and M. Javidi, Numerical approach of the space fractional order diffusion equation based on the third kind of Chebyshev polynomials, Combinatorics, Crytography, Computer Sci. and Computing, Conference paper, 2019.
  • [4]          J. J. Ahmed, Designing the shape of coronal virus using the PDE method, General Letters in Math., 8(2) (2020), 75–82.
  • [5]          H. Ahmad, A. Akgül, T. A. Khan, P. S. Stanimirovi, and Y. Chu, New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations, Complexity, 2020 (2020), 8829017.
  • [6]          D. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403–1412.
  • [7]          J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. Mineola, NY, USA: Dover, 2001.
  • [8]          B. A. Carreras, V. E. Lynch, and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle travers in plasma turbulence models, Phys. Plasma, 8(8) (2001), 5096–5103.
  • [9]          M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 21 (2010), 1021–1032.
  • [10]        M. Dehghan and A. Saadatmandi, Chebyshev finite difference method for Fredholm integro-differential equation, Int. J. Comput. Math., 85 (2008) 123–130.
  • [11]        K. Diethelm,The Analysis of Fractional Differential Equations, Berlin, Germany: Springer-Verlag, 2010.
  • [12]        E. H. Doha, A. H. Bhrawy and S. S. Ezz-Eldien, A new Jacobi operational matrix: An application for solving fractional differential equations, J. Applied Mathematical Modelling, 36 (2012), 4931–4943.
  • [13]        H. Engler, Similarity solutions for a class of hyperbolic integrodifferential equations, Diff. Int. Equations, 10(5) (1997), 815–840.
  • [14]        V. J. Ervin and J. P. Roop, Variational solution  of fractional advection dispersion equations on bounded  domains  in Rd, Numer. Meth. Part D, 23 (2007), 256–281.
  • [15]        Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27 (1990), 309–321.
  • [16]        R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Frac. Calc. Appl. Anal., 1(2) (1998), 167–191.
  • [17]        R. M. Hafez and Y. H. Youssri, Shifted Gegenbauer-Gauss collocation method for solving fractional neutral functional-differential equations with proportional delays, Kragujevac Journal of Mathematics, 46(6) (2022), 981– 996.
  • [18]        M. M. Izadkhah and J. Saberi-Nadjafi, Gegenbauer spectral method for time-fractional convection-diffusion equa- tions with variable coefficients, Mathematical Methods in the Applied Sc., 38(15) (2015), 3183–3194.
  • [19]        H. Jaleb and H. Adibi, On a novel modification of the Legendre collocation method for solving fractional diffusion equation, Computational Methods for Differential Equations, 10(2) (2019), 480–496.
  • [20]        J. T. Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16(3) (2011), 1140–1153.
  • [21]        S. T. Maliheh and S. Elyas, Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations, J. Mathematical Sciences, 00351-8 (2020). DOI: 020-00351-8
  • [22]        J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall, CRC Press, 2003.
  • [23]        M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.
  • [24]        P. Pandey and J. F. Gómez-Aguilar, On solution of a class of nonlinear variable order fractional reaction-diffusion equation with Mittag-Leffler kernel, Numer. Methods Partial Differential Eq., 22563 (2020), 1–14.
  • [25]        I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering. London, UK, 1999.
  • [26]        Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechan- ics:novel trends and recent result, Appl. Mech. Rev., 63 (2010), 010801-1.
  • [27]        A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Computers & Math. with Applic., 62 (2011), 1135–1142.
  • [28]        N. H. Sweilam, A. M. Nagy, and 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), 1283–1297.
  • [29]        N. H. Sweilam, A. M. Nagy, and A. A. El-Sayed, Second kind shifted Chebyshev polynomials for solving space fractional order diffusion equation, Chaos, Solitons & Fractals, 75 (2015), 141 - 147.
  • [30]        G. Szegö, Orthogonal Polynomials, 4th Ed., AMS Colloq. Publ., 1975.
  • [31]        A. Viguerie, G. Lorenzo,F. Auricchio,D. Baroli, T. J. R. Hughes, A. Patton, A. Reali, T. E. Yankeelov, and 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.
  • [32]        H. Wang and N. Yamamoto, Using a partial differential equation with Google  mobility data to predict  COVID-19  in Arizona. Mathematical Biosciences and Engineering., 17(5) (2020), 4891–4904.
  • [33]        Y. H. Youssri, W. M. Abd-Elhameed, and E. H. Doha, Accurate spectral solutions of first- and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method, Applications and Applied Mathematics, 7(3) (2015), 835–851.
  • [34]        Yu Luchko, M. Rivero, J. J. Trujillo, and M. P. Velasco, Fractional models,non-locality, and complex systems, Comput. Math. Appl., 59(3) (2010), 1048–1056.
  • [35]        M. A. Zaky, A. S. Hendy, and J. E. Macás-Díaz, Semi-implicit Galerkin–Legendre Spectral Schemes for Non- linear Time-Space Fractional Diffusion–Reaction Equations with Smooth and Nonsmooth Solutions, J. Scientific Computing, 82(13) (2020), 1-27.