Changpin Li

Hengfei Ding, Changpin Li

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order $O(τ^2+h^2)$, where $τ$ and $h$ are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.

Hengfei Ding, Changpin Li

It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the nonlocal properties of fractional operators. Therefore, developing some high-order numerical approximation formulas for fractional derivatives play a more important role in numerically solving fractional differential equations. This paper focuses on constructing (generalized) high-order fractional-compact numerical approximation formulas for Riesz derivatives. Then we apply the developed formulas to the one- and two-dimension Riesz spatial fractional reaction-dispersion equations. The stability and convergence of the derived numerical algorithms are strictly studied by using the energy analysis method. Finally, numerical simulations are given to demonstrate the efficiency and convergence orders of the presented numerical algorithms.

Hengfei Ding, Changpin Li

Compared to the the classical first-order Grünwald-Letnikov formula at time $t_{k+1} (\textmd{or}\, t_{k})$, 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 $$ \begin{array}{lll} \displaystyle \,_{\mathrm{RL}}{\mathrm{D}}_{0,t}^αu\left(t\right)\left|\right._{t=t_{k+\frac{1}{2}}}= τ^{-α}\sum\limits_{\ell=0}^{k} \varpi_{\ell}^{(α)}u\left(t_k-\ellτ\right) +\mathcal{O}(τ^2),\,\,k=0,1,\ldots, α\in(0,1), \end{array} $$ where the coefficients $\varpi_{\ell}^{(α)}$ $(\ell=0,1,\ldots,k)$ can be determined via the following generating function $$ \begin{array}{lll} \displaystyle G(z)=\left(\frac{3α+1}{2α}-\frac{2α+1}αz+\frac{α+1}{2α}z^2\right)^α,\;|z|<1. \end{array} $$ 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 $\mathcal{O}(τ^2+h^4)$ and $\mathcal{O}(τ^2+h_x^4+h_y^4)$ are shown, where $τ$ 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.

Fanhai Zeng, Changpin Li

In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices are investigated numerically. The spectral collocation schemes are illustrated to solve the fractional ordinary differential equations and fractional partial differential equations. Numerical examples are also presented to illustrate the effectiveness of the derived methods, which show better performances over some existing methods.

Hengfei Ding, Changpin Li

In this paper, we establish even order compact numerical schemes (4th-order, 6th-order, 8th-order, 10th-order) for Riesz derivatives by using the symmetrical fractional centred difference operator. Then we apply the derived 4th-order algorithm to the Riesz spatial telegraph equation. We carefully study the stability and convergence by matrix method, and show that convergence orders in temporal and spatial directions are both 4th order. Numerical experiments are displayed which support the compact difference schemes for Riesz derivatives and the Riesz spatial telegraph equation.

Hengfei Ding, Changpin Li

In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference operator. We apply these methods to fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by Fourier method in details. The convergence orders of these numerical schemes are $\mathcal {O}(τ^2+h^6)$ and $\mathcal {O}(τ^2+h^8)$, respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.

Jianxiong Cao, YangQuan Chen, Changpin Li

This paper presents a UAV-based optimal crop-dusting method to control anomalously diffusing infestation of crops. Two anomalous diffusion models are considered, which are, respectively, time-fractional order diffusion equation and space-fractional order diffusion equation. Our problem formulation is motivated by real-time pest management by using networked unmanned cropdusters where the pest spreading is modeled as a fractional diffusion equation. We attempt to solve the optimal dynamic location of actuators by using Centroidal Voronoi Tessellations. A new simulation platform (FO-DiffMAS-2D) for measurement scheduling and controls in fractional order distributed parameter systems is also introduced in this paper. Simulation results are presented to show the effectiveness of the proposed method as well as the role of fractional order in the overall control performance.

Hengfei Ding, Changpin Li

Numerical methods for fractional calculus attract increasing interests due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz space fractional turbulent diffusion equation. Numerical experiments are displayed which support the theoretical analysis.