广义分数阶反应扩散方程的解

On Fractional Diffusion-Advection-Reaction Equation In $\mathbb{R}$

We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are underst…

分数阶反应扩散方程的H-函数解

Solutions of fractional reaction-diffusion equations in terms of the H-function

研究了一类以Caputo导数为时间导数，Riesz-Feller分数阶导数为空间导数的统一分数阶反应扩散方程的解.该解是通过应用拉普拉斯变换和傅里叶变换以H函数的封闭形式导出的。所得结果具有一般性，包括许多作者早期研究的结果，特别是Mainardi等人（2001，2005）对时空分数阶扩散方程基本解的研究，以及Saxena等人（2006 a，b）对分数阶反应扩散方程的研究。利用Riesz-Feller导数的优点在于，含有Riesz-Feller导数的分数阶反应扩散方程的解包含了时空分数阶扩散的基本解，而时空分数阶扩散本身就是中性分数阶扩散、空间分数阶扩散和时间分数阶扩散的推广。这些特殊类型的扩散可以被解释为随时间演化的空间密度函数，并且可以用紧凑形式的H函数表示。

On the Skewed Fractional Diffusion Advection Reaction Equation on the Interval

This article provides techniques of raising the regularity of fractional order equations and resolves fundamental questions on the one-dimensional homogeneous boundary-value problem of skewed (double-sided) fractional diffusion advection reaction equation (FDARE) with variable coefficients on the bounded interval. The existence of the true (classical) solution together with norm estimation is established and the precise regularity bound is found; also, the structure of the solution is unraveled, capturing the essence of regularity, singularity, and other features of the solution. The key analy…

分数阶反应扩散方程的H-函数的进一步解

Further solutions of fractional reaction-diffusion equations in terms of the H-function

本文是我们早先的文章的继续，在这篇文章中，我们导出了与Caputo导数作为时间导数和Riesz-Feller分数阶导数作为空间导数相关联的统一分数阶反应扩散方程的解。本文考虑了一类以Riemann-Liouville分数阶导数为时间导数，Riesz-Feller导数为空间导数的统一反应扩散方程.该解决方案是通过应用拉普拉斯和傅立叶变换在一个紧凑的和封闭的形式的H-函数。得出的结果具有一般性，包括Kilbas等人（2006年a）、Saxena等人（2006年c）和Mathai等人（2010年）早先研究的结果。主要结果以定理的形式给出。一些有趣的特殊情况下的定理也作为推论。

一类分数阶动力学方程和一个分数阶扩散方程的解

Solutions of certain fractional kinetic equations and a fractional diffusion equation

鉴于动力学方程在某些物理问题中的有用性和重要性，本文导出了一个具有一般性质的分数阶动力学方程的显式解，统一和推广了以前的结果.在此基础上，给出了一种求分数阶扩散方程解的较短方法。

Anisotropic solutions of the time-fractional diffusion equation in multiple dimensions

Anomalous diffusion phenomena are ubiquitous in complex media, such as biological tissues. A wide class of sub-diffusive phenomena phenomena is described by the time-fractional diffusion equation. The paper investigates the case of anisotropic fractional diffusion in the Euclidean space. The solution of the fractional sub-diffusion equation can be expressed in terms of the Wright function and its spatial derivatives, parametrized by the directional unit vector (or alternatively a normal hyperplane). Moreover, the multidimensional case could be expressed as a transformation of the one-dimension…

一类具不同分数阶扩散的偏微分方程组温和解的爆破

Blow up of mild solutions of a system of partial differential equations with distinct fractional diffusions

本文给出了一类具有不同分数阶扩散的非自治弱耦合系统的初值问题正温和解爆破的一个充分条件。证明是基于一个特殊的常微分方程系统的爆破研究。

扩散张量图像配准中引入分数阶正则项的变分模型

A variational model with fractional-order regularization term arising in registration of diffusion tensor image

提出了一种新的扩散张量图像配准的分数阶正则化变分模型。并证明了其解的存在性，从而保证了该模型存在正则解。