Numerical treatment of a two-dimensional variable-order fractional nonlinear reaction-diffusion model

A two-dimensional variable-order fractional nonlinear reaction-diffusion model is considered. A second-order spatial accurate semi-implicit alternating direction method for a two-dimensional variable-order fractional nonlinear reaction-diffusion model is proposed. Stability and convergence of the semi-implicit alternating direct method are established. Finally, some numerical examples are given to support our theoretical analysis. These numerical techniques can be used to simulate a two-dimensional variable order fractional FitzHugh-Nagumo model in a rectangular domain. This type of model can be used to describe how electrical currents flow through the heart, controlling its contractions, and are used to ascertain the effects of certain drugs designed to treat arrhythmia.

[1]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[2]  Vicente Grau,et al.  Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization , 2014, Journal of The Royal Society Interface.

[3]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[4]  Fawang Liu,et al.  An implicit numerical method for the two-dimensional fractional percolation equation , 2013, Appl. Math. Comput..

[5]  Thomas W. Martinek,et al.  Electroviscous Fluids. I. Rheological Properties , 1967 .

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

[7]  T. Hartley,et al.  Initialization, conceptualization, and application in the generalized (fractional) calculus. , 2007, Critical reviews in biomedical engineering.

[8]  L. M. Pismen.  Patterns and Interfaces in Dissipative Dynamics , 2009, Encyclopedia of Complexity and Systems Science.

[9]  T F Nonnenmacher,et al.  A fractional calculus approach to self-similar protein dynamics. , 1995, Biophysical journal.

[10]  Mihály Kovács,et al.  Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels , 2007, Bulletin of mathematical biology.

[11]  Jose Alvarez-Ramirez,et al.  A fractional-order Darcy's law , 2007 .

[12]  K. Burrage,et al.  The Systems Biology Approach to Drug Development: Application to Toxicity Assessment of Cardiac Drugs , 2010, Clinical pharmacology and therapeutics.

[13]  José M. Angulo,et al.  Fractional Generalized Random Fields of Variable Order , 2004 .

[14]  Fawang Liu,et al.  Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation , 2009, Appl. Math. Comput..

[15]  Fawang Liu,et al.  Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term , 2013 .

[16]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[17]  Delfim F. M. Torres,et al.  Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term , 2013, 1305.1859.

[18]  D. Benson,et al.  Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications , 2009 .

[19]  Kevin Burrage,et al.  Fractional models for the migration of biological cells in complex spatial domains , 2013 .

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

[21]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

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

[23]  I. Turner,et al.  A novel numerical approximation for the space fractional advection-dispersion equation , 2014 .

[24]  J. Álvarez-Ramírez,et al.  Effective medium equations for fractional Fick's law in porous media , 2007 .

[25]  Mihály Kovács,et al.  Numerical solutions for fractional reaction-diffusion equations , 2008, Comput. Math. Appl..

[26]  Fawang Liu,et al.  A numerical method for the fractional Fitzhugh–Nagumo monodomain model , 2013 .

[27]  Hans-Gerd Leopold,et al.  Embedding of function spaces of variable order of differentiation in function spaces of variable order of integration , 1999 .

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

[29]  C. Luo,et al.  A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. , 1991, Circulation research.