Mittag-Leffler-Pade approximations for the numerical solution of space and time fractional diffusion equations

Abstract

original work is properly cited. AbstractAnomalous diffusion and non-exponential relaxation patterns can be described by a space -time fractional diffusion equation. This paper aims to present a Padé approximation for Mittag-Leffler function mixed finite difference method to develop a numerical method to obtain an approximate solution for the space and time fractional diffusion equation. The truncation error of the method is theoretically analyzed. It is proved that the numerical proposed method is unconditionally stable from the matrix analysis point of view. Finally, some numerical results are given, which demonstrate the efficiency of the approximate scheme.

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

“…Fourth order global Padé approximants (ν = 4) correspond to the types (m, n) with m + n = 9. They include the types (5,4), (6,3), and (7,2). As discussed in subsection 2.2, the approximation R m,n α,β for ν = 4 takes the form…”

“…For E α (−x), x > 0, Starovoitov and Starovoitova [23] analyzed Padé type approximants of the form p n /q m , m ≤ n, and discussed their asymptotic rate of convergence on the compact unit disk as n → ∞. Borhanifar and Valizadeh [3] constructed a fourth order Padé approximant and used it to develop a numerical scheme for the time-space diffusion equation. Iyiola et al [15] constructed a second order non-Padé type rational approximation for E α,β using real distinct poles (RDP).…”

“…Iyiola et al [15] constructed a second order non-Padé type rational approximation for E α,β using real distinct poles (RDP). However, although the approximants in [3] and [15] might be adequate for small values, they fail to account for the asymptotic power law behavior.…”

“…Comparison of the fourth order approximants; R 5,4 α,β , R 6,3 α,β , R 7,2 α,β for β = 1 Comparison of the fourth order approximants; R 5,4 1,β , R 6,3 1,β , R 7,2 1,β for β = 1.01 (left) and β = 2 (right) Comparison of the fourth order approximants; R 5,4 α,α , R 6,3 α,α , R 7,2 α,α for α = 0.5 (left) and α = 0.75 (right) Plots of approximation errors e m,n α,β (x) for the fourth order approximants (7,2), (6,3),(5,4) …”

Compact ADI method for two-dimensional Riesz space fractional diffusion equation

“…Alikhanov [1] constructed a widespread difference approximation of the Caputo fractional derivative for the time fractional diffusion equation with variable coefficients. Borhanifar and Valizadeh [4] considered Mittag-Leffler-Padé approximations for space and time fractional diffusion equations by using shifted Gr ünwald estimate in space, rational recurrence formula in time, and discussed their stabilities and truncation errors. C ¸elik and Duman [7] used the fractional centered difference that introduced by Ortigueira [23] to solve the Riesz fractional diffusion equation and also for this type equation, some of the authors employed the matrix stemming from the discretization of the Riesz space derivative by compact difference scheme and parameter spline function [38] and fractional centered difference formula [26].…”