Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions

Abbaszadeh, 2020, Second-order finite difference/spectral element formulation for solving the fractional advection-diffusion equation, Commun. Appl. Math. Comput. Sci., 1 Abbaszadeh, 2020, A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation, Appl. Numer. Math., 158, 271, 10.1016/j.apnum.2020.07.020 Abbaszadeh, 2021, Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation, Appl. Math. Comput., 392, 125 Abo-Gabal, 2020, On Romanovski–Jacobi polynomials and their related approximation results, Numer. Methods Partial Differ. Equ., 36, 1982, 10.1002/num.22513 Alikhanov, 2015, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424, 10.1016/j.jcp.2014.09.031 Benson, 2000, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36, 1413, 10.1029/2000WR900032 Bhrawy, 2016, Shifted fractional-order jacobi orthogonal functions: application to a system of fractional differential equations, Appl. Math. Model., 40, 832, 10.1016/j.apm.2015.06.012 Bhrawy, 2017, An improved collocation method for multi-dimensional space–time variable-order fractional schrödinger equations, Appl. Numer. Math., 111, 197, 10.1016/j.apnum.2016.09.009 Canuto, 2007 Chen, 2016, Generalized jacobi functions and their applications to fractional differential equations, Math. Comput., 85, 1603, 10.1090/mcom3035 Dehghan, 2016, Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation, Appl. Math. Model., 40, 3635, 10.1016/j.apm.2015.10.036 Deng, 2009, Finite element method for the space and time fractional Fokker–Planck equation, SIAM J. Numer. Anal., 47, 204, 10.1137/080714130 Doha, 2004, On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. Phys. A, Math. Gen., 37, 657, 10.1088/0305-4470/37/3/010 Hendy, 2020, Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations, Appl. Numer. Math., 156, 276, 10.1016/j.apnum.2020.05.002 Hesthaven, 2007 Heydari, 2020, Orthonormal shifted discrete Legendre polynomials for solving a coupled system of nonlinear variable-order time fractional reaction-advection-diffusion equations, Appl. Numer. Math. Hou, 2018, Müntz spectral methods for the time-fractional diffusion equation, Comput. Methods Appl. Math., 18, 43, 10.1515/cmam-2017-0027 Jin, 2013, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51, 445, 10.1137/120873984 Jin, 2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36, 197 Jin, 2019, Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview, Comput. Methods Appl. Mech. Eng., 346, 332, 10.1016/j.cma.2018.12.011 Liu, 2019, A new spectral method using nonstandard singular basis functions for time-fractional differential equations, Commun. Appl. Math. Comput. Sci., 1, 207, 10.1007/s42967-019-00012-1 Lui, 2020, Spectral collocation in space and time for linear PDEs, J. Comput. Phys., 424 Lyu, 2020, A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equation with nonsmooth solutions, Numer. Methods Partial Differ. Equ., 36, 579, 10.1002/num.22441 Masjedjamei, 2002, Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, Integral Transforms Spec. Funct., 13, 169, 10.1080/10652460212898 Nikan, 2020, Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics, Eng. Anal. Bound. Elem., 120, 223, 10.1016/j.enganabound.2020.08.017 Podlubny, 1998 Sakamoto, 2011, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426, 10.1016/j.jmaa.2011.04.058 Shen, 2011 Sheng, 2020, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal., 58, 2435, 10.1137/19M128377X Stynes, 2016, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal., 19, 1554, 10.1515/fca-2016-0080 Sun, 2006, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56, 193, 10.1016/j.apnum.2005.03.003 Sun, 2018, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64, 213, 10.1016/j.cnsns.2018.04.019 Tarasov, 2020 Tian, 2014, Polynomial spectral collocation method for space fractional advection–diffusion equation, Numer. Methods Partial Differ. Equ., 30, 514, 10.1002/num.21822 Yang, 2020, An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains, J. Comput. Phys., 408, 10.1016/j.jcp.2020.109284 Zaky, 2019, Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions, J. Comput. Appl. Math., 357, 103, 10.1016/j.cam.2019.01.046 Zaky, 2019, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math., 145, 429, 10.1016/j.apnum.2019.05.008 Zaky, 2020, An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions, Appl. Numer. Math., 154, 205, 10.1016/j.apnum.2020.04.002 Zaky, 2020, A priori error estimates of a jacobi spectral method for nonlinear systems of fractional boundary value problems and related volterra-fredholm integral equations with smooth solutions, Numer. Algorithms, 84, 63, 10.1007/s11075-019-00743-5 Zaky, 2020, Convergence analysis of an L 1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations, Int. J. Comput. Math., 1 Zaky, 2021, An efficient dissipation–preserving legendre–galerkin spectral method for the higgs boson equation in the de sitter spacetime universe, Appl. Numer. Math., 160, 281, 10.1016/j.apnum.2020.10.013 Zaky, 2020, Multi-dimensional spectral tau methods for distributed-order fractional diffusion equations, Comput. Math. Appl., 79, 476, 10.1016/j.camwa.2019.07.008 Zaky, 2020, Semi-implicit Galerkin–Legendre spectral schemes for nonlinear time-space fractional diffusion–reaction equations with smooth and nonsmooth solutions, J. Sci. Comput., 82, 1, 10.1007/s10915-019-01117-8 Zaky, 2021, A unified spectral collocation method for nonlinear systems of multi-dimensional integral equations with convergence analysis, Appl. Numer. Math., 161, 27, 10.1016/j.apnum.2020.10.028 Zaslavsky, 2002, Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461, 10.1016/S0370-1573(02)00331-9 Zayernouri, 2013, Fractional Sturm–Liouville eigen-problems: theory and numerical approximation, J. Comput. Phys., 252, 495, 10.1016/j.jcp.2013.06.031 Zayernouri, 2015, A unified Petrov–Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Eng., 283, 1545, 10.1016/j.cma.2014.10.051 Zheng, 2020, A Legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions, Appl. Math. Lett., 104, 10.1016/j.aml.2020.106247 Zhou, 2020, Nonuniform alikhanov linearized galerkin finite element methods for nonlinear time-fractional parabolic equations, J. Sci. Comput., 85, 1, 10.1007/s10915-020-01350-6