Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method

In this paper, we consider linear and non‐linear space–time fractional reaction–diffusion equations (STFRDE) on a finite domain. The equations are obtained from standard reaction–diffusion equations by replacing a second‐order space derivative by a fractional derivative of order β∈(1, 2], and a first‐order time derivative by a fractional derivative of order α∈(0, 1]. We use the Adomian decomposition method to construct explicit solutions of the linear and non‐linear STFRDE. Finally, some examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Fokker Planck equation solved in terms of complex fractional moments

In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones, the solution of the FPK equation in terms of a finite number of complex moments may be easily found. It is shown that the probability density function (PDF) solution of the FPK equation is restored in the whole domain, including the trend at infinity with the exception of the value of the PDF in zero.     

Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

Some efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an $L1$ approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.     

具有分数阶形状记忆合金系统的时滞反馈控制与稳定性分析

将分数阶引入到形状记忆合金振子模型中,针对分数阶对其系统动力学的影响进行了研究。首先基于分数阶微分方程理论,构建了分数阶形状记忆合金系统,并给出了该系统的稳定性和Hopf分岔存在性条件。其次,设计了时滞反馈控制器,用来控制分数阶形状记忆合金系统的稳定性。研究结果表明,形状记忆合金系统中的时滞和分数阶对系统的动力学性质有着重要的调控作用。     

分数阶退化时滞微分系统的稳定性问题

本文主要研究了含有退化和时滞因素的分数阶微分系统的稳定性．首先,利用分数阶Laplace变换讨论了在Caputo分数阶导数意义下的具有不同分数阶导数的多时滞退化微分系统的稳定性条件,并且对于不同的情况给出了相应的判别方法．其次,利用特征方程的根的分析结果研究了分数阶退化时滞微分系统的全时滞稳定性问题,给出了判定该系统全时滞稳定的充分性条件,获得的结果推广了相关文献的结论．最后给出了具体例子说明了定理条件的有效性．     

非广延等离子体中非线性离子声波的数值研究：分数阶模型（英）

本文的主要目的是研究非磁化等离子体中离子声波的非局部非线性行为,该非磁化等离子体是由正离子、满足非广延分布的电子、以及带有负电荷的静止的尘埃颗粒构成的．在流体力学基本方程组中,本文引入修正的Riemann-Liouville分数阶导数并建立了分数阶模型,结合约化摄动法推导出描述离子声波运动的Korteweg　de Vries (Kdv)方程．本文采用Chebyshev-Legendre-Galerkin (CLG)拟谱方法数值求解该方程,并分析等离子体参数对离子孤立声波结构的影响．本文的研究结果表明：提高分数阶导数的阶数能够提升孤立波的振幅．该结果将有助于更好地理解天体物理和实验室等离子体中的非线性波动现象．     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.     

On the Fractional Order Model of Viscoelasticity

Fractional order models of viscoelasticity have proven to be very useful for modeling of polymers. Time domain responses as stress relaxation and creep as well as frequency domain responses are well represented. The drawback of fractional order models is that the fractional order operators are difficult to handle numerically. This is in particular true for fractional derivative operators. Here we propose a formulation based on internal variables of stress kind. The corresponding rate equations then involves a fractional integral which means that they can be identified as Volterra integral equations of the second kind. The kernel of a fractional integral is integrable and positive definite. By using this, we show that a unique solution exists to the rate equation. A motivation for using fractional operators in viscoelasticity is that a whole spectrum of damping mechanisms can be included in a single internal variable. This is further motivated here. By a suitable choice of material parameters for the classical viscoelastic model, we observe both numerically and analytically that the classical model with a large number of internal variables (each representing a specific damping mechanism) converges to the fractional order model with a single internal variable. Finally, we show that the fractional order viscoelastic model satisfies the Clausius–Duhem inequality (CDI).     

一类特殊形式的时间分数阶Navier-Stokes方程的解

本文考虑了一类特殊形式的时间分数阶Navier-Stokes方程的解,采用分离变量法对方程进行变量分离,得到仅关于空间变量和仅关于时间变量的两个方程,前者是一奇异的Sturm-Liouville问题,利用Bessel函数求解.后者则是一个关于时间的分数阶常微分方程,分别采用积分变换、算子方法和Adomian分解法对其求解,得到的解一致.     

Fractional derivatives and recurrent neural networks in rheological modelling – part I: theory

The aim of this paper was to develop a general approach based on fractional time derivatives and recurrent neural networks to model the rheological behaviour of asphalt materials. The paper focuses on elastic and viscoelastic material characteristics. It consists of two parts. In this first part, the theoretical aspects of modelling are discussed. A brief introduction into the theory of rheological elements based on fractional time derivatives is provided. The fractional differential equation of a general rheological element (base element) is developed from which a huge variety of other rheological elements can be derived, e.g. fractional Newton, Kelvin and standard solid elements. A new approach is presented for solving the fractional differential equations. Artificial neural networks are developed to compute the stress–strain–time behaviour of fractional rheological elements in a numerical efficient way. The approach is tested and verified. The second part of this work will appear later. It will be focused on applications of the new theoretical work to pavement engineering problems.     

Time‐dependent fractional advection–diffusion equations by an implicit MLS meshless method

Recently, many new applications in engineering and science are governed by a series of fractional partial differential equations (FPDEs). Unlike the normal partial differential equations (PDEs), the differential order in an FPDE is with a fractional order, which will lead to new challenges for numerical simulation, because most existing numerical simulation techniques are developed for the PDE with an integer differential order. The current dominant numerical method for FPDEs is finite difference method (FDM), which is usually difficult to handle a complex problem domain, and also difficult to use irregular nodal distribution. This paper aims to develop an implicit meshless approach based on the moving least squares (MLS) approximation for numerical simulation of fractional advection–diffusion equations (FADE), which is a typical FPDE The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless strong‐forms. The stability and convergence related to the time discretization of this approach are then discussed and theoretically proven. Several numerical examples with different problem domains and different nodal distributions are used to validate and investigate the accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the FADE. Copyright © 2011 John Wiley & Sons, Ltd.     

Poisson white noise parametric input and response by using complex fractional moments

In this paper the solution of the generalization of the Kolmogorov–Feller equation to the case of parametric input is treated. The solution is obtained by using complex Mellin transform and complex fractional moments. Applying an invertible nonlinear transformation, it is possible to convert the original system into an artificial one driven by an external Poisson white noise process. Then, the problem of finding the evolution of the probability density function (PDF) for nonlinear systems driven by parametric non-normal white noise process may be addressed in determining the PDF evolution of a corresponding artificial system with external type of loading.     

On the Laplace-transformed-based local meshless method for fractional-order diffusion equation

In this work, we formulate a local meshless method based on Laplace transform to estimate the solution of a time-fractional diffusion equation. The collocation is constructed over small subdomains and combined with Laplace transform for a temporal variable. In this approach, the differentiation matrices are constructed by solving small systems over small local domains instead of a large global collocation matrix. The application of Laplace transform avoids the classical time-stepping procedure. This method is capable of solving fractional differential equations in multidimensions with higher accuracy.     

Numerical inversions for diffusion coefficients in two-dimensional space fractional diffusion equation

This article deals with an inverse problem of determining the diffusion coefficients in 2D fractional diffusion equation with a Dirichlet boundary condition by the final observations at the final time. The forward problem is solved by the alternating direction implicit finite-difference scheme with the discrete of fractional derivative by shift Grünwald formula and a numerical text which is to prove its numerically stability and convergence is given. Furthermore, the homotopy regularization algorithm with the regularization parameter chosen by a Sigmoid-type function is introduced to solve the inversion problem numerically. Numerical inversions both with accurate data and noisy data are carried out for the unknown diffusion coefficients of constant and variable with polynomials, trigonometric and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining diffusion coefficients in 2D space fractional diffusion equation, and the algorithm is also numerically stable for additional date having random noises.     

Modeling of materials with fading memory using neural networks

A neural network‐based concept for the solution of a fractional differential equation is presented in this paper. Fractional differential equations are used to model the behavior of rheological materials that exhibit special load (stress) history characteristics (e.g. fading memory). The new concept focuses on rheological materials that exhibit Newtonian‐like displacement behavior when undergoing (time varying) creep loads. For this purpose, a partial recurrent artificial neural network is developed. The network supersedes the storage of the entire load (stress) history in contrast to the exact solution of the fractional differential equation, where access to all previous load (stress) increments is required to determine the new displacement (strain) increment. The network is trained using data obtained from six different creep simulations. These creep simulations have been conducted by means of the exact solution of the fractional differential equation, which is also included in the paper. Furthermore, the network architecture as well as a complete set of network parameters is given. A validation of the network has been carried out and its outcome is discussed in the paper. To illustrate the particular way the network works, all relevant algorithms (e.g. scaling of the input data, data processing, transformation of the output signal, etc.) are provided to the reader in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.     

分数阶微分双参数黏弹性地基矩形板动力响应

基于弹性地基Pasternak双参数模型,利用分数阶微分得到黏弹性地基双参数模型,并在此基础上建立采用分数阶微分Kelvin模型的双参数黏弹性地基上弹性和黏弹性矩形板在动荷载作用下的动力方程;利用Galerkin方法和分段处理的数值计算方法求解四边简支的弹性和黏弹性地基板的动力方程,通过自由振动算例验证该求解方法的正确性;并分析冲击动荷载作用下分数阶微分Kelvin模型的分数阶、粘滞系数、水平剪切系数和模量参数对位移响应的影响。结果表明：分数阶微分黏弹性模型可以描述不同黏弹性材料的力学行为;分数阶取值0.5前后,矩形板位移响应值出现了不同的衰减发展形态;粘滞系数、水平剪切系数和模量系数取值越大,位移响应衰减速度越快。     

分数阶脉冲微分方程初值问题解的存在性

利用Green函数可以将分数阶微分方程初值问题转化为等价的积分方程.近来此方法被应用于讨论非线性分数阶微分方程初值问题解的存在性.本文讨论菲线性分数阶脉冲微分方程初值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Carathéodory条件,利用非紧性测度的性质和M(o)nch,8不动点定理证明解的存在性.     

Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions

Two higher-order fractional viscoelastic material models consisting of the fractional Voigt model (FVM) and the fractional Maxwell model (FMM) are considered. Their higher-order fractional constitutive equations are derived due to the models’ constructions. We call them the higher-order fractional constitutive equations because they contain three different fractional parameters and the maximum order of equations is more than one. The relaxation and creep functions of the higher-order fractional constitutive equations are obtained by Laplace transform method. As particular cases, the analytical solutions of standard (integer-order) quadratic constitutive equations are contained. The generalized Mittag–Leffler function and H-Fox function play an important role in the solutions of the higher-order fractional constitutive equations. Finally, experimental data of human cranial bone are used to fit with the models given by this paper. The fitting plots show that the models given in the paper are efficient in describing the property of viscoelastic materials.     

Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations

A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete $l^2$ norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.