变阶分数阶微分方程的数值解法综述

近年来,分数阶微积分作为一种工具已经被广泛应用于工程中的各个领域.较常阶分数阶微积分算子而言,变阶分数阶微积分算子能够更加准确地描述复杂系统的物理特性,变阶分数阶微积分建模作为一个强大的数学工具,为工程建模提供了便利.在前人优秀研究成果的基础上,结合近几年的国内外相关学者的研究成果对变分数阶微积分方程的研究作全面的综述.以变阶分数阶微分方程、变阶时间分数阶对流-扩散方程、变阶分数阶反应-扩散方程、变阶分数阶积分-微分方程和时滞变阶分数阶微分方程为主要研究对象,从变分数阶微积分算子的相关定义、模型、数值解及在工程中的相关应用等几个方面进行介绍.研究发现,近年来的算法多集中在多项式算法的基础上,通过构建不同的运算矩阵来实现变阶微分方程到代数方程组的转换.该综述可为相关领域的研究学者提供参考.     

豪斯道夫微积分和分数阶微积分模型的分形分析

清晰解读豪斯道夫微积分和分数阶微积分阶数的分形维意义,并比较这2种微积分建模方法的区别与联系.这是首次清晰定量地导出分数阶微积分的分形几何基础.提供豪斯道夫导数模型描述历史依赖过程的几何解释,即初始时刻依赖性问题,并与分数阶导数模型对比.基于本文作者的早期工作,详细描述非欧几里得距离的豪斯道夫分形距离定义——豪斯道夫导数扩散方程的基本解就是基于该豪斯道夫分形距离.该基本解实质上就是目前广泛使用的伸展高斯分布和伸展指数衰减统计模型.     

分形微积分算子的定义及其应用

基于隐式微积分建模方法,提出分形维空间基本解的概念,从而定义分形维上的微积分算子,用以描述分形材料的各种力学行为.分形微积分算子极大地推广经典的连续介质力学微积分建模方法的使用范围,是分形导数概念的进一步发展.运用奇异边界法成功地数值模拟分形维拉普拉斯算子方程唯象描述的分形材料势问题.     

无刷直流电机的分数阶建模方法

传统的基于机理的整数阶无刷直流电机仿真模型和实际物理样机有较大差距.论文通过分析无刷直流电机的数学模型,在基于机理建模整数阶模型基础上引入分数阶微积分,将机理建模和数字建模相结合,提出一种新的无刷直流电机建模方法.首先根据物理样机建立基于机理的无刷直流电机仿真模型,然后采用基于输出误差的最小二乘法辨识出分数阶模型的参数...     

分数阶系统的分数阶PID控制器设计

对于一些复杂的实际系统,用分数阶微积分方程建模要比整数阶模型更简洁准确.分数阶微积分也为描述动态过程提供了一个很好的工具.对于分数阶模型需要提出相应的分数阶控制器来提高控制效果.本文针对分数阶受控对象,提出了一种分数阶PID控制器的设计方法.并用具体实例演示了对于分数阶系统模型,采用分数阶控制器比采用古典的PID控制器取得更好的效果.     

密度感知质心的3D打印平衡性优化建模

密度是几何模型最重要的物理属性之一,可以决定模型的质量、质心以及转动惯量等物理属性.为实现单一材质下模型的质心与质量的建模,提出一种密度感知质心的内部支撑结构建模方法.首先采用隐式曲面建模的方法来实现模型的偏置,利用体素化的算法对偏置模型进行空间域的几何分割;然后构建目标函数方程以及约束条件并进行求解;最后采用隐函数构造体积分数可控的多孔结构,实现空间域的密度建模.实验结果表明,该方法能够构建密度可控的内部支撑结构来满足质心、质量等物理属性的几何建模,并且能够保证内部结构之间的平滑连接.     

基于最优Oustaloup的分数阶PID参数整定

目前工程控制中大部分系统采用传统PID控制,由于分数阶PID继承了传统PID的优点,并且具有更好的控制品质及更强的鲁棒性,因此针对分数阶微积分的高精度数字实现及分数阶PID控制器在工程复杂系统中的实际应用,提出一种新的分数阶微积分高精度数字实现算法-最优Oustaloup数字实现,并建立控制系统的仿真模型,利用框图式模型结合最优ITAE性能指标来整定分数阶PID的参数。通过实例仿真验证,该方法能进一步优化控制器参数,提高控制精度及获得更好的控制效果,便于非线性系统及复杂系统的分数阶PID参数整定。     

Haar小波求解线性分数阶微积分方程

分数阶微积分方程的应用逐渐受到广大研究者的重视.论文以Haar小波基函数来逼近线性分数阶微积分方程,通过数值算例的数值解与精确解进行比较,结果表明论文方法是有效的且具有较高的精度.     

计及温控负荷响应的二维云模型分布式频率控制

为改善电力系统频率稳定性,充分调用需求侧可控负荷资源,本文提出一种计及温控负荷响应的二维云模型分布式频率控制方法。建立了多区域互联电力系统负荷频率控制模型,设计了基于福克普朗克方程的温控负荷分布式控制策略,同时采用云模型算法与分数阶微积分理论,设计了二维云模型分数阶PID分布式频率控制器。最后通过控制仿真比较与分析,验证了在不同运行场景下所提出的综合控制方法具有较优的动稳态性能。结果表明该控制方法是可行和有效的。     

一类不确定分数阶混沌系统同步的自适应滑模控制方法

针对一类系统不确定及受外界干扰的分数阶混沌系统,本文首先将分数阶微积分应用到滑模控制中,构造了一个具有分数阶积分项的滑模面.针对系统不确定及外界干扰项,基于分数阶Lyapunov稳定性理论与自适应控制方法,设计了一种滑模控制器以及分数阶次的参数自适应律,实现了两不确定分数阶混沌系统的同步控制,并辨识出相应误差系统中不确定项及外界干扰项的边界.在分数阶系统稳定性分析中使用的分数阶Lyapunov稳定性理论及相关函数都可以很好地运用到其它分数阶系统同步控制方法中.最后数值仿真验证了所提控制方法的可行性与有效性.     

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.     

Fractional strain energy and its application to the free vibration analysis of a plate

Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and provides a new approach to non-local mechanics. In this study, a theoretical consideration on a new fractional non-local model is presented based on existence of fractional strain energy. It has two additional free parameters compared to classical local mechanics: (1) a fractional parameter which controls the strain gradient order in the strain energy relation and makes the model more flexible to describe physical phenomena, and (2) a non-local parameter to consider small scale effects in micron and sub-micron scales. The model has been used to obtain a fractional non-local plate theory. Free vibrations of a rectangular simply supported (S–S–S–S) plate has been investigated and the meaning of different parameters, such as fractional and non-local parameters, has been shown. The non-linear governing equation has been solved by the Galerkin method. A simple form of the governing equation and the numerical solution is an advantage of this fractional non-local model. Furthermore, the fractional nonlocal theory is contrasted with the Eringen nonlocal theory to show that fractional one enables to obtain much wider class of solutions.

     

A new operational matrix for solving fractional-order differential equations

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. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. 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 greatly simplifying 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 presented technique.     

Conservation laws of (3+α)-dimensional time-fractional diffusion equation

The concept of Lie–Backlund symmetry plays a fundamental role in applied mathematics. It is clear that in order to find conservation laws for a given partial differential equations (PDEs) or fractional differential equations (FDEs) by using Lagrangian function, firstly, we need to obtain the symmetries of the considered equation.Fractional derivation is an efficient tool for interpretation of mathematical methods. Many applications of fractional calculus can be found in various fields of sciences as physics (classic, quantum mechanics and thermodynamics), biology, economics, engineering and etc. So in this paper, we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations. In the sequel, we obtain our results in order to find conservation laws of the time-fractional equation in some special cases.     

Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation

In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation. This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributed-order Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the Grunwald-Letnikov method (GLM) and the fractional differential transform method (FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.      

分数阶微积分流变模型在岩体结构加速流变破坏分析中的应用

针对采用常用的元件模型较好地拟合试验结果时需要的参数较多的缺点,将分数阶微积分理论应用于岩体结构的流变分析.将分数阶微积分的黏弹性模型和黏塑性模型的一维本构关系推广为三维本构关系,推导流变应变增量计算公式.然后将分数阶微积分流变模型应用于不同岩体结构的加速流变破坏分析.结果表明：改变分数阶次时,岩体结构加速流变性态将发生较大变化,分数阶微积分流变模型可较好地描述岩体结构不同的加速流变破坏过程,且模型简单实用.     

A finite difference method with non-uniform timesteps for fractional diffusion equations

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behavior of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.