Quasi-wavelet method for time-dependent fractional partial differential equation

In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential equations. The time variable is discretized by using the second order backward differentiation formula scheme, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. The comparisons of the present numerical results with the exact analytical solutions show that the quasi-wavelet method has distinctive local property and can achieve accurate results.

[1]  Da Xu,et al.  Quasi wavelet based numerical method for a class of partial integro-differential equation , 2012, Appl. Math. Comput..

[2]  G. Wei.  Quasi wavelets and quasi interpolating wavelets , 1998 .

[3]  G. Wei,et al.  A unified approach for the solution of the Fokker-Planck equation , 2000, physics/0004074.

[4]  G. Wei,et al.  Discrete singular convolution for the solution of the Fokker – Planck equation , 1999 .

[5]  Yang Xiang,et al.  A NOVEL APPROACH FOR THE ANALYSIS OF HIGH-FREQUENCY VIBRATIONS , 2002 .

[6]  Shan Zhao,et al.  Local spectral time-domain method for electromagnetic wave propagation. , 2003, Optics letters.

[7]  Yang Xiang,et al.  Plate vibration under irregular internal supports , 2002 .

[8]  Yingjun Jiang,et al.  Moving collocation methods for time fractional differential equations and simulation of blowup , 2011 .

[9]  Da Xu,et al.  Quasi-wavelet based numerical method for fourth-order partial integro-differential equations with a weakly singular kernel , 2011, Int. J. Comput. Math..

[10]  Jingtang Ma,et al.  High-order finite element methods for time-fractional partial differential equations , 2011, J. Comput. Appl. Math..

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

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

[13]  Yanping Lin,et al.  Numerical Solutions for second-kind Volterra integral equations by Galerkin methods , 2000 .

[14]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[15]  Shan Zhao,et al.  Numerical solution of the Helmholtz equation with high wavenumbers , 2004 .

[16]  Shan Zhao,et al.  DSC time-domain solution of Maxwell's equations , 2003 .

[17]  Guo-Wei Wei,et al.  Discrete Singular Convolution-Finite Subdomain Method for the Solution of Incompressible Viscous Flows , 2002 .

[18]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[19]  Liwen Qian,et al.  A Note on Regularized Shannon's Sampling Formulae , 2000 .

[20]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[21]  Fawang Liu,et al.  Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term , 2009, J. Comput. Appl. Math..

[22]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

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

[24]  S. Mallat.  Multiresolution approximations and wavelet orthonormal bases of L^2(R) , 1989 .

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

[26]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[27]  Shan Zhao,et al.  Comparison of the Discrete Singular Convolution and Three Other Numerical Schemes for Solving Fisher's Equation , 2003, SIAM J. Sci. Comput..

[28]  A. Zients.  Andy , 2003 .

[29]  Guo-Wei Wei,et al.  Lagrange Distributed Approximating Functionals , 1997 .

[30]  Katja Lindenberg,et al.  Reaction front in an A+B-->C reaction-subdiffusion process. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Y. Xiang,et al.  DSC analysis of free-edged beams by an iteratively matched boundary method , 2005 .

[32]  Da Xu,et al.  Crank-Nicolson/quasi-wavelets method for solving fourth order partial integro-differential equation with a weakly singular kernel , 2013, J. Comput. Phys..

[33]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[34]  Liwen Qian.  On the regularized Whittaker-Kotel'nikov-Shannon sampling formula , 2002 .