Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

Abstract

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides the fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply the fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first-and secondorder accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of the fast algorithms.

Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

“…We analyze the error bound of the fast timestepping FEM for solving (1.1). The basic idea for fast calculating the discrete convolution n j=0 ω (α) n−j u j is to reexpress the convolution weights ω (α) n as an integral, see, e.g., [4,28,41,32]. In [28,41], ω (α) n is expressed into a contour integral, which can be discretized by the exponentially convergent mid-point rule based on the Talbot, parabolic, or hyperbolic contour; see, e.g., [35].…”

“…The direct computation of the above discrete convolution is costly, requiring O(n T ) active memory and O(n 2 T ) operations. The computational difficulty can be resolved by developing fast memory-saving algorithms, see, e.g., [3,4,10,15,16,28,32,41,43].…”

“…Our proof is based on the convergence of the direct computational method, which is simpler than that of the existing fast methods; see, e.g., [32]. To the best of authors' knowledge, this is the first work that unifies the convergence analysis of the popular (fast) time-stepping numerical schemes for solving (1.1), including the fractional backward difference type methods of order one and two [29,34], the fractional Crank-Nicolson type methods [17,40], and the recently developed BN-θ method [39]; see Section 4.…”

A variety of exact solutions for fractional (2+1)-dimensional Heisenberg ferromagnetic spin chain in the semi classical limit

“…D α t (Ψ) is the conformable fraction derivative of Ψ of order α. Nowadays, the fiel of conformable fractional derivative become one of the most important and interesting fiel for scientists because of its uses nonlinear sciences suck as, flui mechanics, chemical and biological processes. In literature, there are so many definition which of them are, Riemann-Liouville [30,31], Atangana-Baleanu derivative in Caputo sense [32], Caputoa and Grunwald-Letnikov [33][34][35].…”

Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states

“…The fractional Fokker-Planck equation models the PDF of the position of the particles [6,7]. So far, there are many numerical methods for solving FFPE, such as finite difference method, finite element method, and even the stochastic methods [11,12,16,24,27].…”