分数阶偏微分方程的分数阶特征方法

具有非奇异核的分数微分方程中的维度一致性

Dimensional consistency in fractional differential equations with non singular kernels

本文探讨了用分数导数代替普通时间导数时出现的维度一致性问题，提出通过变量替换确保非奇异核分数微分方程的物理维度一致性，并给出示例。

关于一类具有无穷多个解的分数阶微分方程

On a fractional differential equation with infinitely many solutions

给出了分数阶微分方程$x^{(α)}(t)=g(x(t))$，$t\geq0$的一组限制条件，其中$α\in(0,1)$和$g(0)=0$，使得从$x(0)=0$开始存在无穷多个解.算符$x^{(α)}$是卡普托微分算符。

Lie Symmetry Analysis of Some Conformable Fractional Partial Differential Equations

In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives. The analysis is applied to Korteweg-de Vries, modified Korteweg-de Vries, Burgers, and modified Burgers equations with conformable fractional time and space derivatives. For each equation, all of the vector fields and the Lie symmetries are obtained. Moreover, exact solutions are given to these equations in terms of solutions of ordinary differential equations. In particular, it is shown that the …

二维Caputo分数阶微分算子的变换性质

Transformation Property of the Caputo Fractional Differential Operator in Two Dimensional Space

导出了二维空间中旋转标量函数的Caputo分数阶导数算子的变换性质。变换性质的研究是建立多维空间分数阶微积分公式的基础。分数阶微积分在拉格朗日和哈密顿动力学中的应用依赖于这种变换。给出了一个说明性的例子。

含Riemann-Liouville分数阶导数的分数阶微分方程初值条件的物理解释

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

从粘弹性领域的一系列例子，我们证明，它是可能的属性的物理意义表示的黎曼-刘维分数导数的初始条件，它是可能的，以获得初始值，这样的初始条件通过适当的测量或观察。

Inverse Problems for One Dimentional Conformable Fractional Dirac Type Integro Differential System

In this paper, one dimentional conformable fractional Dirac-type integro differential system is considered. The asymptotic formulae for the solutions, eigenvalues and nodal points are obtained. We investigate the inverse nodal problem and give an effective procedure for solving the inverse nodal problem with respect to given a dense subset of nodal points .

分数阶非线性微分方程的分数阶变分迭代法

Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations

近年来，分数阶微分方程的研究已通过著名的变分迭代法。然而，以往的工作都避免了分数阶导数项，并将其作为一个限制变分来处理。为了克服这些缺点，提出了一种分数阶变分迭代法。拉格朗日乘子可以明确地确定基于分数阶变分理论。

通向分数阶积分、分数阶微分方程的路径及H函数的作用

Pathway to Fractional Integrals, Fractional Differential Equations and the Role of H-function

该研究重新探究实标量情形下的路径模型，揭示其与分数阶积分、微分方程解等多领域的联系，指出H函数是这些关联的共同主线，且路径参数与分数阶数相关。

Fractional differential and integral operations and cumulative processes

In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of traditional fractional calculus for describing complex systems is uncovered. The connection between complex physics with fractional differentiation and integration operations is established.

Symmetry Analysis of Initial and Boundary Value Problems for Fractional Differential Equations in Caputo sense

In this work we study Lie symmetry analysis of initial and boundary value problems for partial differential equations (PDE) with Caputo fractional derivative. We give generalized definition and theorem for the symmetry method for PDE with Caputo fractional derivative, according to Bluman's definition and theorem for the symmetry analysis of PDE system. We investigate the symmetry analysis of initial and boundary value problem for fractional diffusion and the third order fractional partial differential equation (FPDE). Also we give some solutions.

On solutions of linear fractional differential equations and systems thereof

It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling transformations. We then derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We sh…

分数次微分算子外尔定律的一个简短证明

A short proof of Weyl's law for fractional differential operators

本文研究了有限体积$\R^d$的开子集上的一大类微分算子的谱渐近性。这类算子包括Dirichlet Laplacian算子、分数Laplacian算子以及具有非齐次符号的分数微分算子。基于特征值和的精确估计，我们建立了半经典渐近的首项。这推广了拉普拉斯算子的外尔定律。

变系数线性分式微分算子Green函数的显式表示

Explicit Representations of Green's Function for Linear Fractional Differential Operator with Variable Coefficients

给出了具有{ \it变系数}和Riemann-Liouvilles导数的一般线性分数阶微分算子的格林函数的显式表示.我们假设它们的系数在$[0, \infty)$中是连续的。利用格林函数的显式表示，得到了一般型变系数非齐次分数阶微分方程解的显式表示。因此，将以往研究中发展的求解常系数分数阶微分方程的Green函数方法推广到{ \it变系数}分数阶微分方程的情形。

Lizorkin型空间中的p-Adic分析：分数次算子、伪微分方程和Tauberian定理

p-Adic analysis in the Lizorkin type spaces: fractional operators, pseudo-differential equations and Tauberian theorems

本文引入了检验函数和检验分布的p进Lizorkin空间，并在此空间上研究了多维Vladimirov和Taidon分数次算子。由于p -进Lizorkin空间在Vladimirov和Taiffon算子下是不变的，它们可以在与分数算子问题相关的考虑中发挥关键作用。在Lizorkin空间中引入了一类p-adic伪微分算子，并构造了伪微分方程的解。证明了Lizorkin分布中与分数次算子和伪微分算子有关的p-Adic多维Tauberian定理。

分数阶微分形式II

Fractional Differential Forms II

这项工作进一步发展了分数阶微分形式的性质。特别地，分数形式空间的有限维子空间被认为是。定义了内积、Hodge对偶和协变导数.还计算了积分序形式的坐标变换规则。矩阵阶分数阶微积分用于定义矩阵的阶形式。这是通过将矩阵阶导数与外部导数相结合来实现的。给出了矩阵序形式的坐标变换规则和协变导数。证明了Poincare引理对除阶为不可对角化矩阵外的所有阶外分数次微分积分都成立。

A boundary-value problem for a mixed type equation involving hyper-Bessel fractional differential operator and Hilfer's bi-ordinal fractional derivative

In a rectangular domain, a boundary-value problem is considered for a mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. Using the method of separation of variables, Laplace transform, a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial problems for a differential equation with the bi-ordinal Hilfer's derivative and regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting po…

分数阶线性微分方程的Laplace变换方法

The Laplace Transform Method for Linear Differential Equations of the Fractional Order

介绍了一类分数阶微分方程初值问题的拉普拉斯变换解法。该方法基于双参数Mittag-Leffler函数的Laplace变换。为了推广所提出的方法的情况下，所谓的“序列”分数阶微分方程，拉普拉斯变换的“序列”分数阶导数也得到了。此外，还介绍了通过直接代入相应方程来检验候选解的必要工具：Mittag-Leffler函数的分数阶导数和依赖于参数的积分的分数阶微分规则。给出了分数阶格林函数的定义，并给出了它的一些性质，这些性质对于构造分数阶线性微分方程初值问题的解是必要的。给出了一、二、三和四项方程特殊情况下分数阶格林函数的显式表达式，以及任意n项常系数分数阶线性常微分方程的显式表达式.几个不同类型分数阶微分方程的解的例子，包括分数阶扩散-波动方程。参考书目（54项）涵盖了分数阶导数可能应用的许多领域。

分数阶微积分：分数阶积分和微分方程

Fractional Calculus: Integral and Differential Equations of Fractional Order

在Riemann-Liouville分数阶微积分的框架下，引入了分数阶积分和分数阶微分的线性算子。特别注意的是专门用于处理这些运营商的拉普拉斯变换的技术，以方便应用科学家，避免非生产性的泛泛而谈和过度的数学严谨。应用这一技巧，我们将导出最简单的分数阶线性积分和微分方程的解析解。我们显示的基本作用的Mittag-Leffler函数，其属性报告在一个特设附录。讨论的题目是：（a）Riemann-Liouville分数阶微积分的基本原理和Laplace变换的基本公式，（b）Abel型第一类和第二类积分方程，（c）松弛型和振动型分数阶微分方程。

离散分数阶微积分的矩阵方法II：部分分数阶微分方程

Matrix approach to discrete fractional calculus II: partial fractional differential equations

提出了一种新的离散化方法，该方法能方便地离散具有任意实阶导数（所谓分数阶导数）和时滞的偏微分方程，并以各种类型的分数阶扩散方程的数值解为例加以说明.所建议的方法是Podlubny矩阵方法的发展（Fractional Calculus and Applied Analysis，vol.3，no.4，2000，359- 386）。四个例子的数值解的分数扩散方程的各种组合的时间/空间分数导数（整数/整数，分数/整数，整数/分数，分数/分数）相对于时间和空间变量，以说明如何简单和一般的建议的方法。第五个例子说明了该方法同样可以简单地用于含时滞的分数阶微分方程。一组MATLAB例程的方法，以及用于解决的例子的示例代码的实施已经开发。

分数阶Levy过程和分数阶Levy噪声驱动的随机微分方程的随机积分

Stochastic integration for fractional Levy process and stochastic differential equation driven by fractional Levy noise

在[1]给出的平方可积〔的〕纯跳Levy过程的白噪声分析的基础上，将分数阶Levy过程作为Levy过程的广义泛函，定义了由平方可积〔的〕纯跳Levy过程定义的分数阶Levy过程的形式导数为分数阶Levy噪声，进而定义了关于分数阶Levy过程的Skorohod积分.此外，我们还提出了一类分数阶Levy噪声驱动的随机Volterra方程，并研究了其解的存在唯一性;此外，我们还提出了一类分数阶Levy噪声驱动的随机微分方程，并证明了在Lipschtz和线性条件下，存在唯一的随机分布值解.