Finite difference/spectral approximation for a time-space fractional equation on two and three space dimensions

Abstract We consider the numerical method for solving a time–space fractional equation whose solution has a weak initial singularity. L 1 scheme on graded mesh is used to capture the initial weak singularity. It is shown that the method can recover order 2 − α convergence by choosing some appropriate grading parameters, where α ( 0 α 1 ) is the order of temporal Caputo fractional derivative. A fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2D and 3D cases. Stability and convergence of the fully discrete scheme are rigorously established. Numerical results are presented to show the sharpness of the error estimate.

[1]  Santos B. Yuste,et al.  Subdiffusion-limited A+A reactions. , 2001 .

[2]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[3]  G. Karniadakis,et al.  A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations , 2016 .

[4]  Bangti Jin,et al.  An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data , 2015, 1501.00253.

[5]  Guang-an Zou.  Numerical solutions to time-fractional stochastic partial differential equations , 2018, Numerical Algorithms.

[6]  Weihua Deng,et al.  Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations , 2016, SIAM J. Numer. Anal..

[7]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[8]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[9]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[10]  Yingjun Jiang,et al.  Moving finite element methods for time fractional partial differential equations , 2013 .

[11]  Mark M. Meerschaert,et al.  A second-order accurate numerical method for the two-dimensional fractional diffusion equation , 2007, J. Comput. Phys..

[12]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[13]  V. Gafiychuk,et al.  Mathematical modeling of time fractional reaction-diffusion systems , 2008 .

[14]  Zhiping Mao,et al.  Analysis and Approximation of a Fractional Cahn-Hilliard Equation , 2017, SIAM J. Numer. Anal..

[15]  I. Turner,et al.  Time fractional advection-dispersion equation , 2003 .

[16]  Weihua Deng,et al.  Time Discretization of a Tempered Fractional Feynman-Kac Equation with Measure Data , 2018, SIAM J. Numer. Anal..

[17]  Guang-an Zou,et al.  A Galerkin finite element method for time-fractional stochastic heat equation , 2016, Comput. Math. Appl..

[18]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[19]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[20]  I. Turner,et al.  Numerical Approximation of a Fractional-In-Space Diffusion Equation, I , 2005 .

[21]  K. Burrage,et al.  Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain , 2012 .

[22]  Jose L. Gracia,et al.  A Fitted Scheme for a Caputo Initial-Boundary Value Problem , 2018, J. Sci. Comput..

[23]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[24]  Natalia Kopteva,et al.  Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions , 2017, Math. Comput..

[25]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[26]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[27]  Maarten Wyns.  Convergence analysis of the Modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with nonsmooth initial data , 2015 .

[28]  Jincheng Ren,et al.  L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity , 2019, Int. J. Model. Simul. Sci. Comput..

[29]  Hu Chen,et al.  Error Analysis of a Second-Order Method on Fitted Meshes for a Time-Fractional Diffusion Problem , 2018, J. Sci. Comput..

[30]  Yubin Yan,et al.  An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data , 2018, SIAM J. Numer. Anal..

[31]  Jiwei Zhang,et al.  Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations , 2018, SIAM J. Numer. Anal..

[32]  Xindong Zhang,et al.  Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the fractional diffusion-wave equation , 2014, Numerical Algorithms.

[33]  K. Burrage,et al.  Fourier spectral methods for fractional-in-space reaction-diffusion equations , 2014 .

[34]  I. Podlubny.  Fractional differential equations , 1998 .

[35]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.