分数阶广义扰动热波方程的与泛函映射解

研究了一类分数阶广义非线性扰动热波方程.首先在典型分数阶热波方程情形下得到解,接着用泛函分析映射方法,求出了分数阶广义非线性扰动热波方程初始边值问题的任意次近似解析解.最后简述了它的物理意义.求得的近似解析解,弥补了单纯用数值方法得到的模拟解的不足.     

分数阶广义扰动热波方程

研究了一类分数阶广义非线性扰动热波方程.首先用奇异慑动方法,求出了分数阶广义非线性扰动热波方程初始边值问题的任意次近似解析解.然后利用泛函分析不动点定理证明了它的一致有效性,最后简述了它的物理意义.求得的近似解析解,弥补了单纯用数值方法求模拟解的不足.     

分数阶振子方程基于变分迭代的近似解析解序列

在粘弹性介质中的阻尼振动中引入分数阶微分算子,建立分数阶非线性振动方程.使用了分数阶变分迭代法（FVIM）,推导了Lagrange乘子的若干种形式.对线性分数阶阻尼方程,分别对齐次方程和正弦激励力的非齐次方程应用FVIM得到近似解析解序列.以含激励的Bagley-Torvik方程为例,给出不同分数阶次的位移变化曲线.研究了振子运动与方程中分数阶导数阶次的关系,这可由不同分数阶次下记忆性的强弱来解释.计算方法上,与常规的FVIM相比,引入小参数的改进变分迭代法能够大大扩展问题的收敛区段.最后,以一个含分数导数的Van der Pol方程为例说明了FVIM方法解决非线性分数阶微分问题的有效性和便利性.     

一类分数阶非线性波方程的精确解及应用

基于分离变量的思想构造了分数阶非线性波方程含常系数的解的形式.在用待定系数法求解时,根据原方程确定假设解中的待定参数,得到具体解的表达式.利用该方法求解了3个非线性波方程,即分数阶CH(Camassa-Holm)方程、时间分数阶空间五阶Kdv-like方程、分数阶广义Ostrovsky方程.比较简便地得到了这些方程的精确解.文献中关于整数阶非线性波方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.对能够通过待定系数法求出精确解的分数阶微分方程所应满足的条件进行了阐述.     

(2,q)阶分数差分方程的解

首次提出了一种分数阶差分,分数阶和分以及分数阶差分方程的定义,并给出(2,q)阶常系数分数阶差分方程的具体解法.     

功能梯度压电空心圆柱中分数阶热弹导波的频散和衰减特性

基于分数阶热电弹性理论和Legendre多项式方法,构建了功能梯度空心圆柱中导波传播的数学模型.讨论了分数阶次、压电效应、径厚比等对导波传播,特别是对其衰减的影响规律.数值结果表明,压电效应对衰减的影响主要集中在截止频率和突变频率附近,并使得突变频率发生前移;分数阶对热波模态相速度和衰减的影响较大,且热波相速度存在模态交叉,在交叉频率点附近分数阶对相速度的影响相反;热波衰减随着分数阶增大而逐渐减小;第一阶纵向模态衰减受到了压电效应的抑制,其余模态衰减都显著增大,并且电开路受到的影响要比电短路状态大.     

分数(k,q)阶差分方程的解

本文首次提出了一种分数阶差分,分数阶和分以及分数阶差分方程的定义,并利用Z变换理论,给出(k,q)阶常系数分数阶差分方程的具体解法.     

分数阶超Yang族及其超哈密顿结构

通过应用分数阶超迹恒等式以及建立在李超代数上的广义零曲率方程,得到分数阶超Yang族和它的分数阶超哈密顿结构.应用该方法还可以得到其他分数阶超方程族.     

分数阶Fornberg-Whitham方程及其改进方程的变分迭代解

给出分数阶Fornberg Whitham方程(FFW)并把其中非线性项uu_x换为u2u_x后所得的改进Fornberg-Whitham方程的解．使用了分数阶变分迭代法(fractional variational iteration method,FVIM),其中Lagrange乘子由泛函和Laplace变换确定．讨论了分数阶次的数值在两种情况下FFW方程的解,因为确定FFW方程中时间微分的阶次需要比较原方程中含时间的两个微分的阶次．最后,给出两个使用分数阶变分迭代法的算例．算例结果证明了所提方法的有效性     

分数阶热弹理论下重力场对二维纤维增强介质的影响

基于Sherief等提出的分数阶广义热弹性耦合理论,研究了在热冲击作用下二维纤维增强弹性体的热弹性问题.考虑了重力对二维纤维增强线性热弹性各向同性介质的影响,建立了控制方程.运用正则模态法,经过数值计算,对控制方程进行求解,得到了不同分数阶参数和不同重力场下无量纲温度、位移和应力分量的表达式,以图形的方式展示了变量的分布规律并对结果展开了讨论.研究结果为:重力场和分数阶参数对纤维增强介质的位移及应力有着重要的影响,但对温度的影响有限.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

用格子Boltzmann方法模拟非线性热传导方程

对具有非线性源项和非线性扩散项的热传导方程建立格子Boltzmann求解模型.在演化方程中增加了两个关于源项分布函数的微分算子,对演化方程实施Chapman-Enskog展开.通过对演化方程的进一步改进,恢复出具有高阶截断误差的宏观方程.对不同参数选取下的非线性热传导方程进行了数值模拟,数值解与精确解吻合得很好.该模型也可以用于同类型的其他偏微分方程的数值计算中.     

三相滞后对有球形空腔无限介质双温广义热弹性的影响

就各向同性的无限弹性体,具有一个球形空腔时,从双温广义热弹性理论（2TT）角度,研究三相滞后热方程的热弹性相互作用问题．在三相滞后理论中,热传导方程是一个含时间四阶导数的、双曲型的偏微分方程．假设无限介质初始时静止,通过Laplace变换,将基本方程用向量矩阵微分方程的形式表示,然后通过状态空间法求解．将得到的通解应用于特殊问题:空腔边界上承受着热荷载（热冲击和坡型加热）和力学荷载．使用Fourier级数展开技术,实现Laplace变换的求逆．计算了铜类材料物理量的数值解．图形显示,两种模型:带能量耗散的双温Green-Naghdi理论（2TGNIII）和双温3相滞后模型（2T3相）明显不同．还对双温和坡型参数的影响进行了研究．     

An exact methodology for solving nonlinear diffusion equations based on integral transforms

This paper develops a new methodology for the solution of nonlinear diffusion equations. The solution technique is based on integral transforms and leads to exact numerical results. We apply the formal methodology to the problem of one-dimensional transient heat conduction. A new form of the heat equation is developed using a generalized expression for temperature-dependent thermal conductivity, based on a power-series expansion, for the three standard orthogonal coordinate systems. The resulting form of the heat equation suggests that the finite integral transform technique may reduce the dimensionality of the heat equation prior to the initiation of any numerical procedure. An example in a slab with linearly varying thermal conductivity is shown to produce exact results for the temperature distribution.     

考虑非傅立叶效应的瞬态导热过程的波动响应分析

瞬态导热分析需要考虑非傅立叶效应.通过对抛物型及双曲型热传导方程,以及耦合热传导方程后的波动方程的数值求解,研究了非傅立叶效应下导热过程中的波动响应.结果表明,双曲型热传导过程具有明显的波动特性,所引起的波动响应前沿值成倍提高,且呈现显著的跃变行为,而波动峰值外的部位围绕着初始值小幅波动.     

Thermoelasticity that uses fractional heat conduction equation

A survey of nonlocal generalizations of the Fourier law and heat conduction equation is presented. More attention is focused on the heat conduction with time and space fractional derivatives and on the theory of thermal stresses based on this equation.     

在模糊随机因素影响下的多层圆筒结构的稳态热传导问题的区间数解法

在多层圆筒结构稳态热传导分析中,根据给定固体壁两侧表面温度总传热量公式,首先推导出当边界温度为随机变量情况下总传热量函数统计参数的均值和方差;然后推导出在导热系数为模糊数,边界温度为随机数下的总传热量的区间表达式．通过比较可以知道由区间数算法得到的区间最大,由概率统计算法得到的区间最小．并给出了两者的相对误差公式．最后引进粗糙集中的上、下近似集,提出用一个参数来统一定义模糊和随机区间进行稳态结构的热传导分析．     

Explosive-crystallization waves in thin amorphous layers on heat conducting substrates with thermal contact resistance

A model for explosive crystallization in a thin amorphous layer on a heat conducting substrate is presented. Rate equations are used to describe the kinetics of the homogeneous amorphous-crystalline transition. Heat conduction into the substrate and thermal contact resistance at the interface between layer and substrate are taken into account. The whole process is examined as a wave of invariant shape in a moving frame of reference. A coupled system of an integro-differential equation and ordinary differential equations is obtained and solved numerically. The propagation velocity of the wave is obtained as an eigenvalue of the system of equations. Some representative solutions are shown. Crystallization-wave velocities are compared with experimental values for explosive crystallization in germanium. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

The mathematical model of the three‐dimensional semiconductor devices of heat conduction is described by a system of four quasi‐linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection‐dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008