Higher order Grünwald approximations of fractional derivatives and fractional powers of operators

Abstract

We give stability and consistency results for higher order Grünwald-type formulae used in the approximation of solutions to fractionalin-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space.

A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

“…However, for some of the approximations to fractional derivatives it seems strange that more conditions are actually needed even for the shifted Grünwald-Letnikov formula with first order of accuracy (u ∈ L 1 (R) and u ∈ C α+1 (R) [19]); this is because the schemes are derived by the Fourier analysis which uses information of the whole domain. Although Theorem 1 in this paper shows that the conditions, which are less rigorous and easier to judge, are enough for general high order schemes; it still means more boundary conditions are needed for the space fractional diffusion problem (1).…”

“…The quasi-compact approximations corresponding to (15), where (m, n) is taken as (0, 1), (1,2), and (0, 2), respectively, are and…”

“…While combining (1,2) and (1,8) leads to an unconditionally stable fourth order quasi-compact scheme; Figure 2 shows the eigenvalues distribution of the iterative matrix with α = 1.5, N = 100. …”

“…is considered in the domain Ω = (0, 1) 2 and t > 0, 1 < α, β ≤ 2, with the boundary conditions Table 8 The discrete L 2 errors and their convergence rates to Example 3 at t = 1 by using the second order stable CN-pseudo-compact schemes and the fourth order one (1, 2)+ (1,8) for different α with τ = h and τ = h 2 , respectively. Table 9 The discrete L 2 errors and their convergence rates to Example 3 at t = 1 by using the third order stable CN-pseudo-compact schemes for different α with τ = h/20.…”

“…Using the idea of second order central difference formula, a second order discretization for Riemann-Liouville fractional derivative is established in [31], and the scheme is extended to two dimensional two-sided space fractional convection diffusion equation in finite domain in [12]. By assembling the Grünwald difference operators with different weights and shifts, a class of stable second order discretizations for Riemann-Liouville space fractional derivative is developed in [33,17], and an abstract way of discussing the stability and convergence of the discretizations can be seen in [1]; its corresponding third order quasi-compact scheme is given in [37]. Allowing to use the points outside of the domain, a class of second, third and fourth order approximations for Riemann-Liouville space fractional derivatives are derived in [13] by using the weighted and shifted Lubich difference operators.…”