Fractional Differential Equations

An energy conservative difference scheme for the nonlinear fractional Schrödinger equations

A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

In this paper, we propose a numerical method for the solution of the time-fractional nonlinear Schrodinger equation in one and two dimensions which appear in quantum mechanics. In this method we first approximate the time fractional derivative of the mentioned equation by a scheme of order O ( τ 2 − α ) , 0 α 1 then we will use the Kansa approach to approximate the spatial derivatives. The meshless method has already proved successful in standard quantum mechanics as well as for several other engineering and physical problems. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the fractional quantum mechanics. The results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme.

The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics

Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(hk+1+τ2−α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

Variable-coefficient discrete (G′G)-expansion method for nonlinear differential–difference equations☆

In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Zakharov–Kuznetsov equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditional stable and L 2 error estimate for the linear case with the convergence rate through analysis.

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Shaoli Wang

Papers

Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

Variable-coefficient discrete (G′G)-expansion method for nonlinear differential–difference equations☆

Numerical Analysis of an Implicit Fully Discrete Local Discontinuous Galerkin Method for the Fractional Zakharov–Kuznetsov Equation