A shifted fractional-order Hahn functions Tau method for time-fractional PDE with nonsmooth solution

Document Type : Research Article


Department of applied mathematics, Graduate university of advanced technology, Kerman, Iran.


In this paper, a new orthogonal system of nonpolynomial basis functions is introduced and used to solve a class of time-fractional partial differential equations that have nonsmooth solutions. In fact, unlike polynomial bases, such basis functions have singularity and are constructed with a fractional variable change on Hahn polynomials. This feature leads to obtaining more accurate spectral approximations than polynomial bases. The introduced method is a spectral method that uses the operational matrix of fractional order integral of fractional-order shifted Hahn functions and finally converts
the equation into a matrix equation system. In the introduced method, no collocation method has been used, and initial and boundary conditions are applied during the execution of the method. Error and convergence analysis of the numerical method has been investigated in a Sobolev space. Finally, some numerical experiments are considered in the form of tables and figures to demonstrate the accuracy and capability of the proposed method.


Main Subjects

[1] Abbaszadeh, M. and Dehghan, M. Numerical investigation of repro-ducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estima-tion, Appl. Math. Comput. 392 (2021), 125–718.
[2] Abo-Gabal, H., Zaky, M.A. and Doha, E.H. Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions, Appl. Numer. Math. 182 (2022), 214–234.
[3] Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. The fractional-order governing equation of Levy motion, Water Resour. Res. 36 (6) (2000), 1413–1423.
[4] Bhattacharyya, P.K. Distributions Generalized Functions with Applica-tions in Sobolev Spaces, Distributions. de Gruyter, 2012.
[5] Bhrawy, A. and Zaky, M. An improved collocation method for multi-dimensional space-time variable-order fractional Schroedinger equations, Appl. Numer. Math. 111 (2017), 197–218.
[6] Canuto, C., Quarteroni, A., Hussaini, M.Y. and Zang, T.A. Spectral methods; Fundamentals in single domains, Springer Science & Business Media, 2007.
[7] Chen, S., Shen, J. and Wang, L. Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput. 85 (300) (2016), 1603–1638.
[8] Dehghan, M., Abbaszadeh, M. and Mohebbi, A. Legendre spectral ele-ment method for solving time fractional modified anomalous subdiffusion equation, Appl. Math. Model. 40 (5-6) (2016), 3635–3654.
[9] Deng, W. Finite element method for the space and time fractional Fokker–Planck equation, SINUM. 47 (1) (2009), 204–226.
[10] Goertz, R. and Öffner, P. Spectral accuracy for the Hahn polynomials, ArXiv e-prints: arXiv:1609.07291, 2016.
[11] Hendy, A.S. and Zaky, M.A. Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations, Appl. Numer. Math. 156 (2020), 276–302.
[12] Hesthaven, J.S., Gottlieb, S. and Gottlieb, D. Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computa-tional Mathematics, 21. Cambridge University Press, Cambridge, 2007.
[13] Heydari, M., Avazzadeh, Z. and Atangana, A. Orthonormal shifted discrete Legendre polynomials for solving a coupled system of nonlin-ear variable-order time fractional reaction-advection-diffusion equations, Appl. Numer. Math. 161 (2021), 425–436.
[14] Hou, D., Hasan, M.T. and Xu, C. Muntz spectral methods for the time-fractional diffusion equation, Comput. Methods Appl. Math. 18 (1) (2018), 43–62.
[15] Jin, B., Lazarov, R. and Zhou, Z. Error estimates for a semi-discrete finite element method for fractional order parabolic equations, SIAM Journal on Numerical Analysis 51 (1) (2013), 445–466.
[16] Jin, B., Lazarov, R. and Zhou, Z. Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview, Comput. Methods Appl. Mech. Eng. 346 (2019), 332–358.
[17] Karlin, S. and McGregor, J.L. The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961), 33–46.
[18] Kreyszig, E. Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978.
[19] Latifi, S. and Delkhosh, M. Generalized Lagrange Jacobi‐Gauss‐Lobatto vs Jacobi‐Gauss‐Lobatto collocation approximations for solving (2+ 1)-dimensional sine‐Gordon equations, Math. Methods Appl. Sci. 43(4) (2020), 2001–2019.
[20] Lui, S. and Nataj, S. Spectral collocation in space and time for linear PDEs, J. Comput. Phys. 424 (2020), 109–843.
[21] Lyu, P. and Vong, S. A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equa-tion with nonsmooth solutions, Numer. Methods Partial. Differ. Equ. 36 (3) (2020), 579–600.
[22] Nikan, O., Avazzadeh, Z. and Machado, J.T. Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics, Eng. Anal. Bound. Elem. 120 (2020), 223–237.
[23] Parand, K. and Delkhosh, M. Operational matrices to solve nonlinear Riccati differential equations of arbitrary order, St. Petersburg Poly-technical University Journal: Physics and Mathematics 3 (3) (2017), 242–254.
[24] Podlubny, I. Fractional differential equations: an introduction to frac-tional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[25] Saeedi, H. A fractional-order operational method for numerical treat-ment of multi-order fractional partial differential equation with variable coefficients, SeMA J. 75(3) (2018), 421–433.
[26] Saeedi, H. and Chuev, G.N. Triangular functions for operational ma-trix of nonlinear fractional Volterra integral equations, J. Appl. Math. Comput. 49(1-2) (2015), 213–232.
[27] Sakamoto, K. and Yamamoto, M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (1) (2011), 426–447.
[28] Salehi, F., Saeedi, H. and Mohseni Moghadam, M. Discrete Hahn poly-nomials for numerical solution of two-dimensional variable-order frac-tional Rayleigh–Stokes problem, Comput. Appl. Math. 37 (4) (2018), 5274–5292.
[29] Sheng, C., Shen, J., Tang, T., Wang, L. and Yuan, H. Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal. 58 (5) (2020), 2435–2464.
[30] Tarasov, V.E. Mathematical economics: Application of fractional calcu-lus, Mathematics 8(5) (2020), 660.
[31] Yang, Z., Liu, F., Nie, Y. and Turner, I. An unstructured mesh finite dif-ference/finite element method for the three-dimensional time-space frac-tional Bloch-Torrey equations on irregular domains, J. Comput. Phys. 408 (2020), 109–284.
[32] Zaky, M.A. Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with nonsmooth solutions, J. Comput. Appl. Math. 357 (2019), 103–122.
[33] Zaky, M.A. and Ameen, I.G. A priori error estimates of a Jacobi spec-tral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions, Numer. Algorithms 84(1) (2020), 63–89.
[34] Zaky, M.A. and Hendy, A.S. Convergence analysis of an L1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equa-tions, Int. J. Comput. Math. 98(7) (2021), 1420–1437.
[35] Zaky, M.A., Hendy, A.S. and Macías-Díaz, J.E. Semi-implicit Galerkin–Legendre spectral schemes for nonlinear time-space fractional diffusion–reaction equations with smooth and nonsmooth solutions, J. Sci. Comput. 82 (1) No. 13 (2020), 1–27.
[36] Zaslavsky, G.M. Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (6) (2002), 461–580.
[37] Zayernouri, M., Ainsworth, M. and Karniadakis, G.E. A unified Petrov–Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Eng. 283 (2015), 1545–1569.
[38] Zayernouri, M. and Karniadakis, G.E. Fractional Sturm–Liouville eigen-problems: theory and numerical approximation, J. Comput. Phys. 252 (2013), 495–517.