An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations

Abstract An implicit difference scheme with the truncation of order 2 − α ( 0 α 1 ) for time and order 2 for space is considered for the one-dimensional time-fractional Burgers equations. The L 1 -discretization formula of the fractional derivative in the Caputo sense is employed. The second-order spatial derivative is approximated by means of the three-point centered formula and the nonlinear convection term is discretized by the Galerkin method based on piecewise linear test functions. The stability and convergence in the L ∞ norm are proved by the energy method. Meanwhile, a novel iterative algorithm is proposed and implemented to solve the nonlinear systems. Numerical experiment shows that the results are consistent with our theoretical analysis, and the comparison between the proposed iterative algorithm and the existing methods shows the efficiency of our method.

[1]  J. C. López-Marcos.  A difference scheme for a nonlinear partial integrodifferential equation , 1990 .

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

[3]  Tao Tang,et al.  A finite difference scheme for partial integro-differential equations with a weakly singular kernel , 1993 .

[4]  Ting-Zhu Huang,et al.  A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems , 2018 .

[5]  Anatoly A. Alikhanov,et al.  A new difference scheme for the time fractional diffusion equation , 2014, J. Comput. Phys..

[6]  Cui-Cui Ji,et al.  Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation , 2016, J. Sci. Comput..

[7]  Najeeb Alam Khan,et al.  Numerical solutions of time‐fractional Burgers equations: A comparison between generalized differential transformation technique and homotopy perturbation method , 2012 .

[8]  Zhi‐zhong Sun,et al.  Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme , 2017 .

[9]  A. Esen,et al.  Numerical Solution of Time Fractional Burgers Equation by Cubic B-spline Finite Elements , 2016 .

[10]  Bangti Jin,et al.  An analysis of the Crank–Nicolson method for subdiffusion , 2016, 1607.06948.

[11]  Da Xu,et al.  A Compact Difference Scheme for an Evolution Equation with a Weakly Singular Kernel , 2012 .

[12]  Syed Tauseef Mohyud-Din,et al.  Analytical Approach to Space- and Time-Fractional Burgers Equations , 2010 .

[13]  Chen Hong-bin,et al.  A SECOND ORDER FULLY DISCRETE DIFFERENCE SCHEME FOR A NONLINEAR PARTIAL INTEGRO-DIFFERENTIAL EQUATION , 2008 .

[14]  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..

[15]  D. Kaya,et al.  Numerical and exact solutions for time fractional Burgers' equation , 2017 .

[16]  A. Esen,et al.  Numerical solution of time fractional Burgers equation , 2015 .

[17]  Chengjian Zhang,et al.  A linear finite difference scheme for generalized time fractional Burgers equation , 2016 .

[18]  Jun Zhou,et al.  A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation , 2020, Int. J. Comput. Math..

[19]  Zhimin Zhang,et al.  Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations , 2015, 1511.03453.

[20]  Shaher Momani,et al.  Non-perturbative analytical solutions of the space- and time-fractional Burgers equations , 2006 .

[21]  Ya-Nan Zhang,et al.  Error Estimates of Crank-Nicolson-Type Difference Schemes for the Subdiffusion Equation , 2011, SIAM J. Numer. Anal..

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

[23]  Jian-Guo Liu,et al.  Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives , 2016, Journal of Scientific Computing.