Vector Additive Decomposition for 2 D Fractional Diffusion Equation

Abstract. Such physical processes as the diffusion in the environment s with fractal geometry and the particles’ subdiffusion lead to the initia l value problems for the nonlocal fractional order partial differential equations. Th ese equations are the generalization of the classical integer order differential equations. An analytical solution for fractional order differential e quation with the constant coefficients is obtained in [1] by using Laplace-Fourier tra nsform. However, nowadays many of the practical problems are described by the models wi th variable coefficients. In this paper we discuss the numerical vector decomposition m del which is based on a shifted version of usual Grünwald finite-difference appr oximation [2] for the non-local fractional order operators. We prove the unconditional sta bili y of the method for the fractional diffusion equation with Dirichlet boundary con ditions. Moreover, a numerical example using a finite difference algorithm for 2D fractiona l order partial differential equations is also presented and compared with the exact anal ytical solution.


Introduction
In this paper we use the Riemann-Liouville fractional derivative where n is an integer, and the αth order is in the following interval n − 1 < α ≤ n.
Following [3] the case L = 0 of the formula (1) is called the Riemann form and the case L = ∞ is called Liouville form for the fractional derivatives.With the boundary conditions offered below the Riemann and Liouville forms become equivalent.Grunwald-Letnikov formula for solving the one-dimensional diffusion equation is used in [2] by M.Meerschaert et al.According to the author the application of this formula leads to the unstable algorithm.This fact is the reason for the appearance of the important and interesting scientific results for the numerical methods theory to solve the fractional order differential equations [2].Allowing for [2], in the present paper we use the rightshifted Grünwald approximation which is of the following form at 1 < α ≤ 2 where N 1 is a non-negative integer, Γ(p) is the gamma function and h 1 =

Statement of the problem
On finite rectangular domain where We assume that the differential equation ( 2) has a unique and sufficiently smooth solution under the following initial u( ) on the perimeter of the rectangular region Ω with the additional restriction We replace the domain Ω by a discrete domain and define t n = nτ, 0 ≤ t n ≤ T , Function values in the discrete points are written in the following form u(x 1 , x 2 , t n+1 ) = u ij = u n+1 ij .We assume that the solution function u(x 1 , x 2 , t n ) is sufficiently smooth and vanishes on the left and lower boundary of the rectangular region.We define the finite difference operator using the right shifted Grünwald formula of these type where These normalized weights (4) depend on the index k and the order α only and, for A β 2 y n ij , analogously.
Lets consider one of the parallel systems of the equations, for example (6).Using (3), we obtain the algebraic equations' system The boundary conditions provide a required approximation.Writing (10) so that its realization is convenient with the conditions ( 11), (12).On each time step algorithm (13), ( 11), ( 12) is realized by a forward sweep direction due to re-arrangement of gridpoints in the x 1 -direction, thus it is economical.Analogously, in the x 2 -direction.
Lets compose a matrix Q k at the unknown [y 1j0 , . . ., y N1−1j0 ] T , at each fixed value j = j 0 , (j = 1, N 2 − 1), when the upper index is the line number and the lower index is the column number where δ i l is Croneker symbol.In other words, the equations' system (14), ( 12), (13) can be presented as follows The F i k for each x 2 j0 are defined from the last two expressions on the right-hand side of (13).
For the theorem proof we will apply the useful results described below.
where the right side is the absolute convergent row at |z| ≤ 1. Placing z = −1 in (16), we get and hence +∞ k=0 g αk < 0 for all N > 1.According to Greschgorin theorem [11], we have Moreover, the terms Q i i are the centers of circles with radiuses Thus Re λ( A k ) > 1 and the spectral radius of the inverse matrix is less than 1, ρ( A −1 k ) < 1.We obtain the results for y 2k , analogously.Therefore, all problems,defined by ( 10)-( 12), are unconditionally stable.
Example 1.Consider fractional order partial differential equation here 0 < x 1 , x 2 < 1 for 0 ≤ t ≤ T with the known exact solution u = x 2.9 1 x 2.6 2 e −t .
The diffusion coefficients are and the forcing function is The algorithm was implemented using the Mathematica 5.1 compiler on a Dell Pentium PC.
It should be noted that this example problem does not meet the requirement for the commutativity of the operators in (3) which was used to establish the stability of the numerical decomposition model ( 10)-(12).

Conclusions
The offered numerical vector decomposition model yields a numerical solution that is O(τ 2 ) + O(h 1 ) + O(h 2 ) accurate.We emphasize that the vector additive methods do not meet the requirement for the commutativity of the decomposition operators A α 1 and A β 2 .