Implicit compact difference schemes for the fractional cable equation

In this paper, we propose two implicit compact difference schemes for the fractional cable equation. The first scheme is proved to be stable and convergent in l_∞-norm with the convergence order O(τ + h⁴) by the energy method, where new inner products defined in this paper gives great convenience for the theoretical analysis. Numerical experiments are presented to demonstrate the accuracy and effectiveness of the two compact schemes. The computational results show that the two new schemes proposed in this paper are more accurate and effective than the previous.     

An investigation of some nonclassical methods for the numerical approximation of Caputo-type fractional derivatives

Traditional methods for the numerical approximation of fractional derivatives have a number of drawbacks due to the non-local nature of the fractional differential operators. The main problems are the arithmetic complexity and the potentially high memory requirements when they are implemented on a computer. In a recent paper, Yuan and Agrawal have proposed an approach for operators of order α ∈ (0,1) that differs substantially from the standard methods. We extend the method to arbitrary α > 0, , and give an analysis of the main properties of this approach. In particular it turns out that the original algorithm converges rather slowly. Based on our analysis we are able to identify the source of this slow convergence and propose some modifications leading to a much more satisfactory behaviour. Similar results are obtained for a closely related method proposed by Chatterjee. Dedicated to Professor Paul L. Butzer on the occasion of his 80th birthday.     

Numerical simulation for the three-dimension fractional sub-diffusion equation

Fractional sub-diffusion equations have been widely used to model sub-diffusive systems. Most algorithms are designed for one-dimensional problems due to the memory effect in fractional derivative. In this paper, the numerical simulation of the 3D fractional sub-diffusion equation with a time fractional derivative of order $\alpha$ $(0<\alpha <1)$ is considered. A fractional alternating direction implicit scheme (FADIS) is proposed. We prove that FADIS is uniquely solvable, unconditionally stable and convergent in $H^1$ norm by the energy method. A numerical example is given to demonstrate the efficiency of FADIS.     

On the convergence of a conservative numerical scheme for the usual Rosenau-RLW equation

In this paper, we study the initial-boundary value problem of the usual Rosenau-RLW equation by finite difference method. We design a conservative numerical scheme which preserves the original conservative properties for the equation. The scheme is three-level and linear-implicit. The unique solvability of numerical solutions has been shown. Priori estimate and second order convergence of the finite difference approximate solutions are discussed by discrete energy method. Numerical results demonstrate that the scheme is efficient and accurate.     

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation

     

Fast high order difference schemes for the time fractional telegraph equation

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ 2-1_σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order and , respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ 2-1_σ method.     

A compact difference scheme for the fractional diffusion-wave equation

This article is devoted to the study of high order difference methods for the fractional diffusion-wave equation. The time fractional derivatives are described in the Caputo’s sense. A compact difference scheme is presented and analyzed. It is shown that the difference scheme is unconditionally convergent and stable in _L∞$L_{\infty }$-norm. The convergence order is O(τ^3-α+h⁴)$O({\tau }^{3-\alpha }+h^4)$. Two numerical examples are also given to demonstrate the theoretical results.     

N-dimensional fractional Fokker-Planck equation and its solutions for anomalous radial two-phase flow in porous media

In this paper we present a time fractional Fokker-Planck equation (fFPE) for radial two-phase flow of liquid and gas in porous media. The fFPE of order α is solved for both two- and three-dimensional flow patterns using the Laplace transform method. The general solutions of the fFPE for both two- and three- dimensional flows are given as a convolution integral of the input and a kernel in the Laplace domain. Special solutions for a large value and a periodic boundary condition are also given in the time domain when the inverse Laplace transform can be found analytically. The fFPE for two-phase flow in porous media presented in this paper is the first report of its kind.     

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

     

Numerical solution of fractional oscillator equation

We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.     

Homotopy perturbation method for fractional KdV equation

In this paper, the homotopy perturbation method is directly applied to derive approximate solutions of the fractional KdV equation. The results reveal that the proposed method is very effective and simple for solving approximate solutions of fractional differential equations.     

Galerkin-FEM for obtaining the numerical solution of the linear fractional Klein-Gordon equation

In this paper, an efficient numerical method for solving the linear fractional Klein-Gordon equation (LFKGE) is introduced. The proposed method depends on the Galerkin finite element method (GFEM) using quadratic B-spline base functions and replaces the Caputo fractional derivative using $L2$ discretization formula. The introduced technique reduces LFKGE to a system of algebraic equations, which solved using conjugate gradient method. The study the stability analysis to the approximation obtained by the proposed scheme is given. To test the accuracy of the proposed method we evaluated the error norm $L_{2}$. It is shown that the presented scheme is unconditionally stable. Numerical example is given to show the validity and the accuracy of the introduced algorithm.     

A lattice Boltzmann model for the Fokker-Planck equation

In this paper, a lattice Boltzmann model is presented for solving one and two-dimensional Fokker-Planck equations with variable coefficients. In particular, it is efficient to simulate one-dimensional stochastic processes governed by the Fokker-Planck equation. Numerical results agree well with the exact solutions, which indicates that the proposed model is suitable for solving the Fokker-Planck equation.     

Solutions for the fractional Landau-Lifshitz equation

This article considers the dynamic equation of a reduced model for thin-film micromagnetics deduced by A. DeSimone, R.V. Kohn and F. Otto in [A. DeSimone, R.V. Kohn, F. Otto, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math. 55 (2002) 1-53]. To derive the existence of weak solutions under periodical boundary condition, the authors first prove the existence of smooth solutions for the approximating equation, then prove the convergence of the viscosity solution when the viscosity term vanishes, which implies the existence of solutions for the original equation.     

Average implicit linear difference scheme for generalized Rosenau-Burgers equation

In this paper, a linear three-level average implicit finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-Burgers equation is presented. Existence and uniqueness of numerical solutions are discussed. It is proved that the finite difference scheme is convergent in the order of O(τ² + h²) and stable. Numerical simulations show that the method is efficient.     

Stable numerical schemes for a partly convolutional partial integro-differential equation

A model partial integro-differential operator (PIDO) that contains both local and nonlocal diffusion operators is considered in this article. This type of operators come in modeling various scientific and financial engineering problems. In most cases, people use finite difference schemes to generate solutions of such model problems. We compare and analyze stability and accuracy of two such finite difference schemes. We first present a discrete analogue of the PIDO and then approximate the semi-discrete time dependent problem using two different one step methods and show the stability conditions and the accuracy of the schemes. We use the Fourier transforms throughout our analysis.     

Numerical solution of fractional diffusion-wave equation based on fractional multistep method

This paper is devoted to application of fractional multistep method in the numerical solution of fractional diffusion-wave equation. By transforming the diffusion-wave equation into an equivalent integro-differential equation and applying Lubich’s fractional multistep method of second order we obtain a scheme of order O(τ^α+h²)$O({\tau }^{\alpha }+h^2)$ for 1?α?1.71832$1?\alpha ?1.71832$ where α$\alpha$ is the order of temporal derivative and τ$\tau$ and h denote temporal and spatial stepsizes. The solvability, convergence and stability properties of the algorithm are investigated and numerical experiment is carried out to verify the feasibility of the scheme.