Wanrong Cao

A fourth-order approximation of fractional derivatives with its applications

A new fourth-order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted Grunwald formulae combining the compact technique. The properties of proposed fractional difference quotient operator are presented and proved. Then the new approximation formula is applied to solve the space fractional diffusion equations. By the energy method, the proposed quasi-compact difference scheme is proved to be unconditionally stable and convergent in L2 norm for both 1D and 2D cases. Several numerical examples are given to confirm the theoretical results.

SPECTRAL AND DISCONTINUOUS SPECTRAL ELEMENT METHODS FOR FRACTIONAL DELAY EQUATIONS

We first develop a spectrally accurate Petrov--Galerkin spectral method for fractional delay differential equations (FDDEs). This scheme is developed based on a new spectral theory for fractional Sturm--Liouville problems (FSLPs), which has been recently presented in [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517]. Specifically, we obtain solutions to FDDEs in terms of new nonpolynomial basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of the first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of the second kind (FSLP-II). We prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the PG spectral method. In contrast to standard (nondelay) fractional differential equations, the delay character of FDDEs might induce solutions, which are either nonsmooth or piecewise smooth. In order to effectively trea...

A finite difference scheme for semilinear space-fractional diffusion equations with time delay

A linearized quasi-compact finite difference scheme is proposed for semilinear space-fractional diffusion equations with a fixed time delay. The nonlinear source term is discretized and linearized by Taylors expansion to obtain a second-order discretization in time. The space-fractional derivatives are approximated by a weighted shifted Grunwald-Letnikov formula, which is of fourth order approximation under some smoothness assumptions of the exact solution. Under the local Lipschitz conditions, the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method. Numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme.

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order

On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations

0<\beta<1

On exponential mean-square stability of two-step Maruyama methods for stochastic delay differential equations

. From the known structure of the nonsmooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multirate fractional differential systems as well as multiterm fractional differential systems with nonsmooth solutions.

TIME-SPLITTING SCHEMES FOR FRACTIONAL DIFFERENTIAL EQUATIONS I: SMOOTH SOLUTIONS ∗

The asymptotic mean-square stability of two-step Maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay (NSDDEs). Under the one-sided Lipschitz condition and the linear growth condition, it is proved that a family of implicit two-step Maruyama methods can preserve the stability of the analytic solution in mean-square sense. Numerical results for both a nonlinear NSDDE and a system show that the family of two-step Maruyama methods have better stability than previous two-step Maruyama methods.

Maximum norm error estimates of the Crank-Nicolson scheme for solving a linear moving boundary problem

We are concerned with the exponential mean-square stability of two-step Maruyama methods for stochastic differential equations with time delay. We propose a family of schemes and prove that it can maintain the exponential mean-square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation. Numerical results for linear and nonlinear equations show that this family of two-step Maruyama methods exhibits better stability than previous two-step Maruyama methods.

NUMERICAL METHODS FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS VIA THE WONG-ZAKAI APPROXIMATION ∗

We propose three time-splitting schemes for nonlinear time-fractional differential equations with smooth solutions, where the order of the fractional derivative is

A second-order difference scheme for the time fractional substantial diffusion equation

0<\alpha<1