Leilei Wei

A numerical study based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional coupled Schrödinger system

In this paper we develop and analyze an implicit fully discrete local discontinuous Galerkin (LDG) finite element method for solving the time-fractional coupled Schrodinger system. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Through analysis we show that our scheme is unconditionally stable, and the L^2 error estimate has the convergence rate O(h^k^+^1+(@Dt)^2+(@Dt)^@a^2h^k^+^1^2) for the linear case. Extensive numerical results are provided to demonstrate the efficiency and accuracy of the scheme.

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

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fishers equation, which is obtained from the standard one-dimensional Fishers 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.

Exact solutions of fractional heat-like and wave-like equations with variable coefficients

Purpose – This paper aims to apply fractional variational iteration method using Hes polynomials (FVIMHP) to obtain exact solutions for variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions. Design/methodology/approach – The approach is based on FVIMHP. The authors choose as some examples to illustrate the validity and the advantages of the method. Findings – The results reveal that the FVIMHP method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations. Originality/value – The variable-coefficient fractional heat-like and wave-like equations with fractional order initial and boundary conditions are solved first. Illustrative examples are included to demonstrate the validity and applicability of the method.

Homotopy analysis method for space‐time fractional differential equations

Purpose – In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.Design/methodology/approach – The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.Findings – The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.Originality/value – The idea can be used to find the numerical solutions of other fractional differential equations.

Numerical analysis of an implicit fully discrete local discontinuous Galerkin method for the fractional Zakharov–Kuznetsov equation

Abstract 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.