Xindong Zhang

Homotopy analysis method for higher-order fractional integro-differential equations

In this paper, we present the homotopy analysis method (shortly HAM) for obtaining the numerical solutions of higher-order fractional integro-differential equations with boundary conditions. The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions. The comparison of the results obtained by the HAM 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 us with a simple way to adjust and control the convergence region of series solution.

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.

Analysis of a local discontinuous Galerkin method for time‐fractional advection‐diffusion equations

Purpose – The purpose of this paper is to develop a fully discrete local discontinuous Galerkin (LDG) finite element method for solving a time‐fractional advection‐diffusion equation.Design/methodology/approach – The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space.Findings – By choosing the numerical fluxes carefully the authors scheme is proved to be unconditionally stable and gets L2 error estimates of O(hk+1+(Δt)2+(Δt)α/2hk+(1/2)). Finally Numerical examples are performed to illustrate the effectiveness and the accuracy of the method.Originality/value – The proposed method is different from the traditional LDG method, which discretes an equation in spatial direction and couples an ordinary differential equation (ODE) solver, such as Runger‐Kutta method. This fully discrete scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. Numerical examples prove that the authors method is very effective. Th...

Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear time-fractional Tricomi-type equation (TFTTE), which is obtained from the standard one-dimensional linear Tricomi-type equation by replacing the first-order time derivative with a fractional derivative (of order α, with 1u2009<u2009αu2009≤u20092). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(hku2009+u20091u2009+u2009τ2), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective.

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.

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0u2009<u2009αu2009<u20091). The proposed method is based on high-order FEM for spacial discretization and finite difference method for time discretization. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h r+1u2009+u2009τ2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.

Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the fractional diffusion-wave equation

In this paper, we consider the numerical approximation for the fractional diffusion-wave equation. The main purpose of this paper is to solve and analyze this problem by introducing an implicit fully discrete local discontinuous Galerkin method. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. By choosing the numerical fluxes carefully we prove that our scheme is unconditionally stable and get L2 error estimates of O(hk+1+(Δt)2+(Δt)α2hk+1)

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

O(h^{k+1}+(Delta t)^{2}+(Delta t)^{frac {alpha }{2}}h^{k+1})

Homotopy analysis method for space‐time fractional differential equations

. Finally numerical examples are performed to illustrate the efficiency and the accuracy of the method.

A generalized fractional sub-equation method for fractional differential 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.