[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.
Send comment about this article