Time-space fractional stochastic Ginzburg-Landau equation driven by Gaussian white noise

The current article is devoted to the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations. The dependence of the order of time-fractional derivative, the order of the space-fractional derivative, and the regularity of the initial data are revealed. The global existence and uniqueness of the mild solutions for time-space fractional complex Ginzburg-Landau equation driven by Gaussian white noise are established.     

A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate

In this article, we discuss a space-fractional diffusion logistic population model with Caputo fractional derivative and density-dependent dispersal rate. The numerical solution of the problem is obtained by using a finite difference scheme. The consistency and stability of the scheme for our solution to the problem are also discussed. The effect of the density-dependent dispersal rate and order of the space-fractional derivative are analyzed for the population density and expanding front (moving boundary).     

Numerical solution of two-dimensional time fractional cable equation with Mittag-Leffler kernel

The main motive of this article is to study the recently developed Atangana-Baleanu Caputo (ABC) fractional operator that is obtained by replacing the classical singular kernel by Mittag-Leffler kernel in the definition of the fractional differential operator. We investigate a novel numerical method for the nonlinear two-dimensional cable equation in which time-fractional derivative is of Mittag-Leffler kernel type. First, we derive an approximation formula of the fractional-order ABC derivative of a function t^k using a numerical integration scheme. Using this approximation formula and some properties of shifted Legendre polynomials, we derived the operational matrix of ABC derivative. In the author of knowledge, this operational matrix of ABC derivative is derived the first time. We have shown the efficiency of this newly derived operational matrix by taking one example. Then we solved a new class of fractional partial differential equations (FPDEs) by the implementation of this ABC operational matrix. The two-dimensional model of the time-fractional model of the cable equation is solved and investigated by this method. We have shown the effectiveness and validity of our proposed method by giving the solution of some numerical examples of the two-dimensional fractional cable equation. We compare our obtained numerical results with the analytical results, and we conclude that our proposed numerical method is feasible and the accuracy can be seen by error tables. We see that the accuracy is so good. This method will be very useful to investigate a different type of model that have Mittag-Leffler fractional derivative.     

应用(G'/G~2)-展开法求解非线性分数阶微分方程（英文）

In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.     

Stability and Convergence of an Implicit Difference Approximation for the Space Riesz Fractional Reaction-Dispersion Equation

In this paper,we consider a Riesz space-fractional reaction-dispersion equation (RSFRDE).The RSFRDE is obtained from the classical reaction-dispersion equation by replacing the second-order space derivative with a Riesz derivative of orderβ∈(1,2]. We propose an implicit finite difference approximation for RSFRDE.The stability and convergence of the finite difference approximations are analyzed.Numerical results are found in good agreement with the theoretical analysis.     

Riemann—Liouville型分数阶微分方程的微分变换方法

本文在Riemann-Liouville分数阶导数的广义Taylor公式的基础上,建立了求解Riemann-Liouville型分数阶微分方程的微分变换方法.本文所建立的基于Riemann-Liouville分数阶导数微分变换方法给求解Riemann-Liouville分数阶导数的微分方程提供了一种新工具。     

The construction of operational matrix of fractional derivatives using B-spline functions

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. Here we construct the operational matrix of fractional derivative of order α in the Caputo sense using the linear B-spline functions. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus we can solve directly the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the new technique presented in the current paper.     

Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo‐type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long‐term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.     

Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives

In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method.     

Thoughts on modeling complexity

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This simple model has been applied to all manner of phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model beyond that of mimicking simple statistics to include long‐time memory in the dynamics of complex phenomena. We show that complexity can give rise to fractional‐difference stochastic processes whose continuum limit is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. Furthermore, the index of the inverse power‐law spectrum in many complex processes can be related to the fractional derivative index in the fractional Langevin equation. This fractional stochastic model suggests that a scaling process guides the dynamics of many complex phenomena. The alternative to the fractional Langevin equation is a fractional diffusion equation describing the evolution of the probability density for certain kinds of anomalous diffusion. © 2006 Wiley Periodicals, Inc. Complexity 11: 33–43, 2006     

分数积分的一种数值计算方法及其应用

提出了一种只需要存储部分历史数据的分数积分的数值计算方法,并给出了误差估计。这种方法可对包含分数积分和分数导数的积分-微分方程进行较长时间的数值计算,克服了存储全部历史数据的困难,并能对计算误差进行控制。作为应用,给出了具有分数导数型本构关系的粘弹性Timoshenko梁的动力学行为研究的控制方程,利用分离变量法讨论梁在简谐激励作用下的动力响应,然后用新提出的数值方法对控制方程进行数值计算,数值计算结果和理论结果进行了比较,它们比较吻合。     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法

带乘性噪声的空间分数阶随机非线性Schrödinger方程是一类重要的方程,可应用于描述开放非局部量子系统的演化过程.该方程为一个无穷维分数阶随机Hamilton系统,且具有广义多辛结构和质量守恒的性质.针对该方程的广义多辛形式,在空间上采用拟谱方法离散分数阶微分算子,在时间上则采用隐式中点格式,构造出一类保持全局质量的广义多辛格式.对行波解和平面波解等进行数值模拟,结果验证了所构造格式的有效性和保结构性质,时间均方收敛阶约在0.5到1之间.     

An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model

In this paper, the q -homotopy analysis transform method (q -HATM) is applied to find the solution for the fractional Lakshmanan-Porsezian-Daniel (LPD) model. The LPD model is the generalization of the non-linear Schrödinger (NLS) equation. The proposed method is graceful fusions of Laplace transform technique with q -homotopy analysis scheme, and the derivative is considered in Caputo sense. In order to validate and illustrate the efficiency of the proposed method, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured for the three different cases in terms of 3D and contour plots for diverse values of the fractional order. The obtained results confirm that the future method is easy to implement, highly methodical, and very effective to analyse the behaviour of complex non-linear fractional differential equations exist in the connected areas of science and engineering.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions

Fractional Brusselator reaction-diffusion system (BRDS) is used for modeling of specific chemical reaction-diffusion processes. It may be noted that numerous models in nonlinear science are defined by fractional differential equations (FDEs) in which an unknown function appears under the operation of a fractional-order derivative. Even though many researchers have studied the applicability and practicality of this model, the analytical approach of this model is rarely found in the literature. In this investigation, a novel semi-analytical technique called fractional reduced differential transform method (FRDTM) has been applied to solve the present model, which is characterized by the time-fractional derivative (FD). Obtained outcomes are compared with the solution of other existing methods for a particular case. Also, the convergence analysis of this model has been studied here.     

Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives

In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space-fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.     

A fast mass-conserving explicit splitting method for the stochastic space-fractional nonlinear Schrödinger equation with multiplicative noise

A fast mass-conserving explicit splitting method is proposed to solve the stochastic space-fractional nonlinear Schrödinger equation which includes a nonlinear term, a fractional Laplacian term and a multiplicative noise term resulting from the nonlocal property, the random variation of the media and the exterior random disturbance. The method splits the original equation into two sub-equations so that appropriate numerical methods can be applied to each sub-equation. A variety of numerical examples in both one- and multi-dimensional spaces show that the scheme has good mass conservative property, first-order strong convergence in time and high efficiency.     

An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation

Distributed-order differential equations have recently been investigated for complex dynamical systems, which have been used to describe some important physical phenomena. In this paper, a new time distributed-order and two-sided space-fractional advection-dispersion equation is considered. Firstly, we transform the time distributed-order fractional equation into a multi-term time-space fractional partial differential equation by applying numerical integration. Then an implicit numerical method is constructed to solve the multi-term fractional equation. The uniqueness, stability and convergence of the implicit numerical method are proved. Some numerical results are presented to demonstrate the effectiveness of the method. The method and techniques can be extended to other time distributed-order and space-fractional partial differential equations.     

Regularization of a backward problem for composite fractional relaxation equations

A backward problem for composite fractional relaxation equations is considered with Caputo's fractional derivative, which covers as particular case of Basset problem that concerns the unsteady motion of a particle accelerating in a viscous fluid in fluid dynamics. Based on a spectral problem, the representation of solutions is established. Next, we show the maximal regularity for the corresponding initial value problem. Due to the mildly ill-posedness of current backward problem, the fractional Landweber regularization method will be applied to discuss convergence analysis and error estimates.