High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation

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 $\mathcal {O}(\tau^2+h^6)$ and $\mathcal {O}(\tau^2+h^8)$, respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.Comment:

Numerical algorithm based on Adomian decomposition for fractional differential equations

AbstractIn this paper, a novel algorithm based on Adomian decomposition for fractional differential equations is proposed. Comparing the present method with the fractional Adams method, we use this derived computational method to find a smaller “efficient dimension” such that the fractional Lorenz equation is chaotic. We also apply this new method to the time-fractional Burgers equation with initial and boundary value conditions. Numerical results and computer graphics show that the constructed numerical is efficient

Monte Carlo method for parabolic equations involving fractional Laplacian

We apply the Monte Carlo method to solving the Dirichlet problem of linear parabolic equations with fractional Laplacian. This method exploit- s the idea of weak approximation of related stochastic differential equations driven by the symmetric stable L\'evy process with jumps. We utilize the jump- adapted scheme to approximate L\'evy process which gives exact exit time to the boundary. When the solution has low regularity, we establish a numeri- cal scheme by removing the small jumps of the L\'evy process and then show the convergence order. When the solution has higher regularity, we build up a higher-order numerical scheme by replacing small jumps with a simple process and then display the higher convergence order. Finally, numerical experiments including ten- and one hundred-dimensional cases are presented, which confirm the theoretical estimates and show the numerical efficiency of the proposed schemes for high dimensional parabolic equations.Comment: 30pages 2 figure

Synchronization Analysis of Two Coupled Complex Networks with Time Delays

This paper studies the synchronized motions between two complex networks with time delays, which include individual inner synchronization in each network and outer synchronization between two networks. Based on the Lyapunov stability theory and the linear matrix equality (LMI), a synchronous criterion for inner synchronization inside each network is derived. Numerical examples are given which fit the theoretical analysis. In addition, the involved numerical results show that the delays between two networks have little effect on inner synchronization. It is also shown that synchronous motions within each network or between two networks are not enhanced if individual intranetwork connections are allowed