Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACT We propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.

[1]  M. Caputo,et al.  Experimental and theoretical memory diffusion of water in sand , 2003 .

[2]  E. H. Twizell,et al.  On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations , 1993 .

[3]  Mohsen Zayernouri,et al.  Fractional pseudo-spectral methods for distributed-order fractional PDEs , 2018, Int. J. Comput. Math..

[4]  Monica Moroni,et al.  Flux in Porous Media with Memory: Models and Experiments , 2010 .

[5]  Kai Diethelm,et al.  Numerical analysis for distributed-order differential equations , 2009 .

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

[7]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[8]  Zhi-Zhong Sun,et al.  Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations , 2016, J. Sci. Comput..

[9]  Teodor M. Atanackovic,et al.  Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod , 2010, 1005.3379.

[10]  Zhi-Zhong Sun,et al.  Some high-order difference schemes for the distributed-order differential equations , 2015, J. Comput. Phys..

[11]  Fawang Liu,et al.  Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations , 2015 .

[12]  M. Caputo,et al.  A new dissipation model based on memory mechanism , 1971 .

[13]  Fawang Liu,et al.  Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains , 2015, J. Comput. Phys..

[14]  A. Cloot,et al.  A generalised groundwater flow equation using the concept of non-integer order derivatives , 2007 .

[15]  T. Hashida,et al.  MATHEMATICAL MODELING OF ANOMALOUS DIFFUSION IN POROUS MEDIA , 2011 .

[16]  Michele Caputo,et al.  Diffusion with space memory modelled with distributed order space fractional differential equations , 2003 .

[17]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[18]  Ian W. Turner,et al.  GPU Accelerated Algorithms for Computing Matrix Function Vector Products with Applications to Exponential Integrators and Fractional Diffusion , 2016, SIAM J. Sci. Comput..

[19]  Teodor M. Atanackovic,et al.  Distributed-order fractional wave equation on a finite domain: creep and forced oscillations of a rod , 2011 .

[20]  Abdul-Qayyum M. Khaliq,et al.  Linearly implicit predictor-corrector methods for space-fractional reaction-diffusion equations with non-smooth initial data , 2018, Comput. Math. Appl..

[21]  I M Sokolov,et al.  Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.