Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative

The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.

ONE-PARAMETER FINITE DIFFERENCE METHODS AND THEIR ACCELERATED SCHEMES FOR SPACE-FRACTIONAL SINE-GORDON EQUATIONS WITH DISTRIBUTED DELAY

This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,we construct a kind of oneparameter finite difference(OPFD)method.It is shown that,under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation,the preconditioned conjugate gradient(PCG)method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method.For 2D problems,we develop another kind of OPFD method.For such a method,two classes of accelerated schemes are suggested,one is alternative direction implicit(ADI)scheme and the other is ADI-PCG scheme.In particular,we prove that ADI scheme can arrive at second-order accuracy in time and space.With some numerical experiments,the computational effectiveness and accuracy of the methods are further verified.Moreover,for the suggested methods,a numerical comparison in computational efficiency is presented.

Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Discrete Chaos in Fractional Henon Map

In this study, a discrete fractional Henon map is proposed in the Caputo discrete delta’s sense. The results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.

A High Order Formula to Approximate the Caputo Fractional Derivative

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order𝛼for 0<α<1.This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial diff erential equations.In comparison with the previous formulae,the main superiority of the new formula is its order of accuracy which is 4−α,while the order of accuracy of the previous ones is less than 3.It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost.The eff ectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples.Moreover,an application of the new formula in solving some fractional partial diff erential equations is presented by constructing a fi nite diff erence scheme.A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

Square-Root Dynamics of a SIR-Model in Fractional Order

In this paper, we consider an SIR-model for which the interaction term is the square root of the susceptible and infected individuals in the form of fractional order differential equations. First the non-negative solution of the model in fractional order is presented. Then the local stability analysis of the model in fractional order is discussed. Finally, the general solutions are presented and a discrete-time finite difference scheme is constructed using the nonstandard finite difference (NSFD) method. A comparative study of the classical Runge-Kutta method and ODE45 is presented in the case of integer order derivatives. The solutions obtained are presented graphically.

Computational analysis for fractional characterization of coupled convection-diffusion equations arising in MHD fows

The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional derivative concept,while the model is solved via the full-spectral method(FSM)and the semi-spectral scheme(SSS).The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques.The SSS is developed by discretizing the time variable,and the space domain is collocated by using equal points.A detailed comparative analysis is made through graphs for various parameters and tables with existing literature.The contour graphs are made to show the behaviors of the velocity and magnetic fields.The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows,and the concept may be extended for variable order models arising in MHD flows.

High-order numerical methods of fractional-order Stokes' first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractional- order Stokes＇ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes

Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

The space-time fractional advection dispersion equations are linear partial pseudo-differential equations with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. The time fractional order is denoted by β∈ and ?is devoted to the space fractional order. The time fractional advection dispersion equations describe particle motion with memory in time. Space-fractional advection dispersion equations arise when velocity variations are heavy-tailed and describe particle motion that accounts for variation in the flow field over entire system. In this paper, I focus on finding the precise explicit discrete approximate solutions to these models for some values of ?with ?, ?while the Cauchy case as ?and the classical case as ?with ?are studied separately. I compare the numerical results of these models for different values of ?and ?and for some other related changes. The approximate solutions of these models are also discussed as a random walk with or without a memory depending on the value of . Then I prove that the discrete solution in the Fourierlaplace space of theses models converges in distribution to the Fourier-Laplace transform of the corresponding fractional differential equations for all the fractional values of ?and .

Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System

In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional order.To recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized method.Based on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy method.Theoretical results are also confirmed by a numerical example.

基于L2-1_(σ)格式逼近时间分数阶扩散方程的差分方法及其收敛性分析

针对时间分数阶扩散方程,在时间方向上结合L2-1_(σ)格式,空间上采用二阶中心差分方法进行离散,并对离散格式进行了收敛性和稳定性分析,离散格式和分析方法可以很容易推广到空间高维情形。最后,通过数值算例对L2-1_(σ)格式和L1格式进行了误差和收敛阶的对比,显示出L2-1_(σ)格式在时间分数阶导数逼近上的优势。

空间分数阶导数“反常”扩散方程数值算法的比较

分别采用显式差分格式、隐式差分格式以及Crank-Nicholson差分格式数值求解空间分数阶导数,并从局部截断误差、稳定性、计算量三个方面进行比较分析;通过数值算例验证分析结果.

时间分数阶色散方程的有限差分方法

提出求解时间分数阶色散方程的一类隐式差分格式,并证明其无条件稳定性和收敛性,收敛阶为O(τ+h2)。该分数阶色散方程是将一般的色散方程中的时间一阶导数用α(0<α<1)阶导数代替所得到的。数值算例表明本方法是有效的。

两边空间-时间分数阶扩散方程的加权有限差分格式(英文)

对于空间-时间分数阶扩散方程的初边值问题提出了一种加权差分格式.利用能量估计,得到了差分格式的稳定性.然后使用数学归纳法证明了在相同的条件下,所提出的的格式是收敛的.最后通过一个例子说明了所提出的格式是可靠的、有效的.

一类空间分数阶扩散方程系数反问题的数值解

数值求解一类空间分数阶扩散方程系数反问题。利用函数变换,将源项系数反问题转为对应的定解问题,并利用隐式差分格式求解,然后利用数值积分,求得待定系数函数的数值解,并且证明隐式差分格式的绝对稳定性。数值算例表明,该方法具有较高的精度。

两边空间分数阶对流-扩散方程的一种加权显式有限差分方法

考虑两边空间分数阶对流-扩散方程的初边值问题,基于Grünwald公式和移位Grünwald-Letnikov公式,提出一种加权显式有限差分解法.利用傅里叶变换和特征值法,得到差分格式的稳定性.然后使用最大模估计法证明在相同的条件下,所提出的差分格式是收敛的.最后通过数值例子说明所提出的差分格式是可靠和有效的,并对方程的数值解与精确解进行比较,验证了文中的理论结果.

分数阶常微分方程的数值解法

研究分数阶常微分方程,用Grunwald近似逼近分数阶导数,用向后差分逼近一阶导数,构造了差分格式,证明差分格式是稳定的和收敛的,并列举数值例子以说明理论分析是正确的.

二维变系数空间分数阶电报方程数值解

针对二维变系数空间分数阶电报方程,利用Grünwald-Letnikov分数阶导数的定义,在交替方向法的基础上提出了一种分数阶Peaceman-Rachford差分格式.通过Gerschgorin定理和Lax等价定理证明了所提出的分数阶Peaceman-Rachford差分格式是无条件稳定和收敛的.数值试验表明:分数阶Peaceman-Rachford差分格式是有效和可靠的.