Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

Abstract

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy. References D. Baleanu, Z. B. Guvenc, J. A. T. Machado. (Eds.) New Trends in Nanotechnology and Fractional Calculus Applications. Springer, 2010. R. E. Ewing, T. Lin and Y. Lin. On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numerical Analysis 39(6):1865--1888, 2002. doi:10.1137/S0036142900368873 M. Ilic, F. Liu, I. Turner, and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)---with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal., 9:333--349, 2006. http://hdl.handle.net/10525/1287 R. Klages, G. Radons, I. M. Sokolov. (Eds.) Anomalous Transport: Foundations and Applications. Wiley, 2008. Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comp. Phys., 225:1533--1552, 2007. doi:10.1016/j.jcp.2007.02.001 I. Podlubny. Fractional Differential Equations. Academic Press, New York, 1999. Y. Saad. Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 29:209--228, 1992. doi:10.1137/0729014 J. Sabatier, O. P. Agrawal, J. A. T. Machado. (Eds.) Advances in fractional calculus: Theoretical developments and applications in physics and engineering. Springer, 2007. V. Simoncini and D. B. Szyld. Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl., 14:1--59, 2007. doi:10.1002/nla.499 H. A. van der Vorst. An iterative solution method for solving $f(A)x=b$ using Krylov subspace information obtained for the symmetric positive definite matrix $A$. J. Comput. Appl. Math., 18:249--263, 1987. doi:10.1016/0377-0427(87)90020-3 Q. Yang, F. Liu, and I. Turner. Stability and convergence of an effective numerical method for the time-space fractional Fokker--Planck equation with a nonlinear source term. International Journal of Differential Equations, Article ID 464321, 22 pages, 2010. doi:10.1155/2010/464321 Q. Yang, F. Liu, and I. Turner. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling, 34:200--218, 2010. doi:10.1016/j.apm.2009.04.006 Q. Yang, I. Turner, F. Liu, and M. Ilic. Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Scientific Computing, 33:1159--1180, 2011. doi:10.1137/100800634