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

Abstract

Compared to the the classical first-order Grünwald-Letnikov formula at time $t_{k+1}(\text{or}{t}_k)$, we firstly propose a second-order numerical approximate scheme for discretizing the Riemann-Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank-Niclson technique applied to the time-fractional differential equations. The established formula has the following form $$\begin{array}{l}{}_{\mathrm{RL}}{\mathrm{D}}_{0,t}^{\alpha }u\left(t\right){|}_{t=t_{k+\frac{1}{2}}}={\tau }^{-\alpha }\sum _{\ell =0}^k{\varpi }_{\ell }^{(\alpha )}u\left(t_k-\ell \tau \right)+\mathcal{O}({\tau }^2),k=0,1,\dots ,\alpha \in (0,1),\end{array}$$ where the coefficients ${\varpi }_{\ell }^{(\alpha )}$ $(\ell =0,1,\dots ,k)$ can be determined via the following generating function $$\begin{array}{l}G(z)={\left(\frac{3\alpha +1}{2\alpha }-\frac{2\alpha +1}{\alpha }z+\frac{\alpha +1}{2\alpha }{z}^2\right)}^{\alpha },|z|<1.\end{array}$$ Applying this formula to the time fractional Cable equations with Riemann-liouville derivative in one or 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.

arXiv comment33 pages, 4 figures

References in corpus (7)

• A new difference scheme for the time fractional diffusion equation
• On an explicit finite difference method for fractional diffusion equations
• From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion
• Matrix approach to discrete fractional calculus II: partial fractional differential equations
• Fractional calculus and continuous-time finance II: the waiting-time distribution
• Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation
• High-order numerical algorithms for Riesz derivatives via constructing new generating functions

Cited by in corpus (2)

• Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes
• Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions