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.     

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.     

分数阶波方程的数值解法

首先,把分数阶波方程转换成等价的积分-微分方程;然后,利用带权的分数阶矩形公式和紧差分算子分别对时间和空间方向进行离散.证明了当权重为1/2时,时间方向的收敛阶为α,其中α(1α2)为Caputo导数的阶数.利用Gronwall不等式,证明了数值格式的收敛性和稳定性.数值例子进一步表明了数值格式的有效性.     

分数阶及整数阶Klein-Gordon方程的孤立波解研究

Klein-Gordon方程是量子力学领域的一类重要方程,它是薛定谔方程的一种相对论形式,包括分数阶和整数阶方程,寻求它的解有着重要的意义.利用一种较为实用的1/G展开法,对一类分数阶Klein-Gordon方程和相应的整数阶Klein-Gordon方程进行了求解,得到了丰富的行波解,包括孤立波解和扭曲波解,同时有代表性地选择一些解,来画出它们的图形并进行相图分析.另外,对所得到的整数阶与分数阶方程的解进行了对比,发现了它们的异同点.     

有限区间上的分数阶扩散-波方程定解问题与Laplace变换

求解了如下的分数阶扩散-波方程定解问题0Dαtu=2ux2,00,0<α≤2,u(0,t;α)=0,u(1,t;α)=θ(t),u(x,0+;α)=0,当1<α≤2时,还有ut(x,0+;α)=0.其中θ(t)是Heaviside单位阶跃函数,0Dαt为关于时间t的α阶Caputo分数阶导数算子,u=u(x,t;α)为时间t的因果函数(即t<0时恒为零的函数).利用Laplace变换的复围道积分反演和离散化反演及FoxH函数理论,给出在计算上对大的t和小的t分别适用的解的表达式.     

Poisson和公式在分数阶热方程中的应用

本文利用Poisson和公式,证明了如下分数阶热方程(D_t~αlu=D_x~2u u(x1 0)=f(x))当f分别为周期函数和f∈S(■)时(速降函数空间),它们的热核满足关系H_t~α(x)=∑n=-∞H_t~α(x+n)进一步,我们把结论推广到更一般的分数阶微分方程和高维情形     

非线性分数阶微分积分方程多点分数阶边值问题解的存在性与唯一性

本文研究一类非线性分数阶微分积分方程多点分数阶边值问题解的存在性与唯一性,利用一些标准的不动点定理进行证明.     

黏弹性材料本构方程的广义分数阶单元网络表述及其广义解

提出广义分数阶单元网络, 取消了Schiessel等人所提出的分数阶单元法对参数的限制, 增加了"协调方程", 将模型解的构造扩充到广义函数空间, 使其包含更多的具有明显物理意义的解. 应用并发展了离散求逆Laplace变换的方法, 给出了模型方程的广义解. 讨论了广义分数阶单元网络Zener, Poyinting-Thomson模型. 结果表明, 有关黏弹性材料本构方程前人所得的经典整数阶和分数阶单参数模型的所有结果均可作为本文的特例而被包括.     

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.     

(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.     

A new fractional derivative for solving time fractional diffusion wave equation

The time fractional diffusion wave equation, which can be used to describe wave diffusion process in this article, was studied. First of all, the diffusion wave equation can be extended to a generalized form in the sense of the regularized version of the $k$-Hilfer–Prabhakar ( $k$-H-P) fractional operator involving the $k$-Mittag- function. Then, the analytical solution can be obtained for this considered equation by using the Laplace transform method and the Fourier transform method. As a result, a novel and general solution have been found. The unconventional solution may show new result and phenomenon to wave diffusion process. Thereby, this research provides a window for discovering new diffusion mechanisms.     

Analytical solution for a generalized space‐time fractional telegraph equation

In this paper, we consider a nonhomogeneous space‐time fractional telegraph equation defined in a bounded space domain, which is obtained from the standard telegraph equation by replacing the first‐order or second‐order time derivative by the Caputo fractional derivative , α > 0 and the Laplacian operator by the fractional Laplacian ( ? Δ)^{β ∕ 2}, β ∈ (0,2]. We discuss and derive the analytical solutions under nonhomogeneous Dirichlet and Neumann boundary conditions by using the method of separation of variables. The obtained solutions are expressed through multivariate Mittag‐Leffler type functions. Special cases of solutions are also discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

Dual characterization of fractional capacity via solution of fractional p-Laplace equation

In terms of weak solutions of the fractional p-Laplace equation with measure data, this paper offers a dual characterization for the fractional Sobolev capacity on bounded domain. In addition, two further results are given: one is an equivalent estimate for the fractional Sobolev capacity; the other is the removability of sets of zero capacity and its relation to solutions of the fractional p-Laplace equation.     

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t) . Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.     

On the local fractional wave equation in fractal strings

The key aim of the present study is to attain nondifferentiable solutions of extended wave equation by making use of a local fractional derivative describing fractal strings by applying local fractional homotopy perturbation Laplace transform scheme. The convergence and uniqueness of the obtained solution by using suggested scheme is also examined. To determine the computational efficiency of offered scheme, some numerical examples are discussed. The results extracted with the aid of this technique verify that the suggested algorithm is suitable to execute, and numerical computational work is very interesting.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.