A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

Abstract

Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

Convolution Quadrature Methods for Time-Space Fractional Nonlinear Diffusion-Wave Equations

“…Ding et al [9] proposed a second-order midpoint approximation formula for the Riemann-Liouville derivative at time t n+1/2 , which is well suited to the Crank-Nicolson scheme for time fractional differential equations. Aiming to reduce the smoothness requirement of this approximation and to improve the numerical stability, we consider a second-order midpoint approximation formula for Riemann-Liouville integral at time t n−1/2 .…”

The unified theory of shifted convolution quadrature for fractional calculus

“…In this example we consider the generalized BDF2-θ method that generalizes the work of Liu et al [8] and Ding et al [11]. Define the generating function ω(ξ) by…”

“…If we take θ = − 1 2 , we get a SCQ that approximates f at the node x n+ 1 2 (see [11] with α replaced by −α),…”

“…Such kind of superconvergence schemes are important for the numerical analysis for PDEs since the resulted schemes possess some good characters, see [5]. Some other methods such as the shifted Grünwald formula [16], the WSGL operator [6], the method developed by Ding et al [11] and recently proposed higher-order approximation formulas [18] that generalize the fractional BDFp for p = 1, 2, • • • , 6 are excluded from the CQ theory as well, since all above methods approximate I α at a shifted node.…”

Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions

“…From the methods above one can see that the key point of efficiently deriving the numerical solutions is developing novel methods to descretize the fractional derivative of the equation, and theoretically showing that the resulted scheme is stable with a high-order convergence rate. To this end, some high-order approximation formulas were developed for the fractional calculus, see [2,9,[12][13][14]16,27,[35][36][37][38]40]. As is well known that the solutions of fractional PDEs show some singularity at initial value, some methods or techniques were developed to cope with such difficulty, see [2,32,33].…”