Fundamental Solutions and Numerical Methods of the Fractional Advection-Dispersion Equations

分数阶微分方程的特点是含有非整数阶导数，能非常有效的描述各种各样的物质的记忆和遗传性质，在物理，数学，机械工程，生物，电子工程，控制理论和金融等领域发挥越来越重要的作用。各种分数阶模型与无秩序的动力系统有着紧密的联系。物理学中的反常扩散最初是从随机游走模型中发展得来的。分数阶对流-扩散方程是模拟各种反常扩散现象的有力工具。分数阶对流-扩散方程是分数阶动力方程的一部分，方程中可以含有空间和时间的分数阶导数算子。本文分别讨论时间、空间、空间-时间的分数阶对流-扩散方程。文中所涉及的空间分数阶导数均为Riesz空间分数阶导数，它含有双侧的Riemann-Liouville分数阶导数。Riesz空间分...The characteristic of fractional order differential equation is containing the non-integer order derivative. It can effectively describe the memory and transmissibility of many kinds of material, and plays an increasingly important role in physics,mathematics, mechanical engineering, biology, electrical engineering, control theory, finance and other fields. All kinds of fractional models have clos...学位：理学博士院系专业：数学科学学院信息与计算数学系_计算数学学号：2005140300

Numerical Methods and Error Analysis of Fractional Differential Equations

分数阶计算是一门正在兴起的学科,它在很多科学领域中发挥了越来越重要的作用,特别在工程,物理,金融,水文等领域.与整数阶模型相比,分数阶模型的显著优点在于它有着深厚的物理背景.但是,由于缺乏较恰当的数学方法,对分数阶计算的理论分析和数值方法的研究还是比较困难的课题.尽管现在大量的应用科学领域已牵涉到分数阶的微分方程,非常少的文献讨论分数阶微分方程的数值方法,尤其是对分数阶偏微分方程的数值方法的探讨.本文分别考虑了分数阶Bagley-Torvik方程和分数阶偏微分方程问题.第一章，先给出有关分数阶导数的一些预备知识.第二章，考虑分数阶常微分方程---Bagley-Torvik方程，给出其解的存在性...Fractional calculation is a developing science. It plays a more and more important role in various fields of science, especially in engineering, physics, finance, and hydrology. The most significant advantage of the fractional order models in comparison with integer-order models is based on important fundamental physical considerations. However, because of the absence of appropriate mathematical m...学位：理学硕士院系专业：数学系_应用数学学号：20022304

Riesz 空间分数阶对流扩散方程的

Riesz 空间分数阶对流扩散方程是从混沌动力系统导出的. 继续Ilic ,Liu 等的工作,我们提出在有界区域内求解 Riesz 空间分数阶对流2扩散方程的一种新的计算有效方法. 即基于这两个Riesz 空间分数阶导数的矩阵表示. 这个方法 的创新在于这个算子的标准离散得到包含具有相同分数次幂的矩阵的一个常微分方程组,并利用计算有效的分数阶行 方法求解. 同时借助于分数阶导数的谱表示和拉普拉斯变换,导出这个Riesz 空间分数阶对流扩散方程的解析解. 最后 给出了数值例子来证实数值方法的有效性.国家自然科学基金(10271098) ,澳大利亚国家研究基金(LP0348653

A Computationally Efficient Solution Method for a Riesz Space Fractional Advection-Dispersion Equation

Riesz空间分数阶对流扩散方程是从混沌动力系统导出的.继续Ilic,Liu等的工作,我们提出在有界区域内求解Riesz空间分数阶对流-扩散方程的一种新的计算有效方法.即基于这两个Riesz空间分数阶导数的矩阵表示.这个方法的创新在于这个算子的标准离散得到包含具有相同分数次幂的矩阵的一个常微分方程组,并利用计算有效的分数阶行方法求解.同时借助于分数阶导数的谱表示和拉普拉斯变换,导出这个Riesz空间分数阶对流扩散方程的解析解.最后给出了数值例子来证实数值方法的有效性.In this paper,a Riesz space fractional advection-dispersion equation(RSFADE) is considered,which is derived from the kinetics of chaotic dynamics.Following work by Ilic and Liu et al,a new computationally efficient method for solving the RSFADE on a bounded domain is proposed.The method is based on the matrix representation of both the Riesz space fractional operators.The novelty of this method is that a standard discretisation of the operator leads to a system of ordinary differential equations(ODEs) with the matrix raised the same fractional power.Then the ODEs is solved by a computationally efficient fractional method of lines.Using a spectral representation of the fractional derivatives and the Laplace transform,the analysis solution of this equation is also derived.Finally,a numerical example is given to demonstrate that this numerical method is computationally efficient.国家自然科学基金(10271098);; 澳大利亚国家研究基金(LP0348653)资

A Computationally Effective Numerical Method for the Fractional-order Bagley-Torvik Equation

给出分数阶Bagley Torvik方程解的存在性和惟一性,导出了用格林函数表示分数阶Bagley Torvik方程的解析解.利用Riemann Liouville定义和Gr櫣nwald Letnikov定义之间的关系,提出了求解分数阶Bagley Torvik方程的一种更有效的数值方法.最后给出数值例子.In this paper, the existence and uniqueness of solution for the fractional-order Bagley-Torvik equation is given. The analytical solution of fractional-order Bagley-Torvik equation is derived by the corresponding Green's function.Using the relationship between the Riemann-Liouville definition and the Grünwald-Letnikov definition, a computationally effective method is proposed for the fractional-order Bagley-Torvik Equation.Finally , some numerical examples are given.国家自然科学基金(10271098)资

A Computationally Effective Numerical Method for the Fractional2order Bagley2Torvik Equation

摘要: 给出分数阶Bagley2Torvik 方程解的存在性和惟一性,导出了用格林函数表示分数阶Bagley2Torvik 方程的解 析解. 利用Riemann2Liouville 定义和Grünwald2Letnikov 定义之间的关系,提出了求解分数阶Bagley2Torvik 方程的一 种更有效的数值方法. 最后给出数值例子 Abstract : In this paper , the existence and uniqueness of solution for the f ractional2order Bagley2Torvik equa2 tion is given. The analytical solution of f ractional2order Bagley2Torvik equation is derived by the corresponding Green’s function. Using the relationship between the Riemann2Liouville definition and the Grünwald2Letnikov definition , a computationally effective method is proposed for the f ractional2order Bagley2Torvik Equation. Final2 ly , some numerical examples are given.国家自然科学基金(10271098) 资

Fractional Cable Equation by Finite Volume Method

CAblE方程是神经元动力学中最重要的基本方程之一,而用于描述神经纤维活动的分数阶CAblE方程能够更好地模拟神经元的动力学行为.文章采用有限体积法离散得到数值逼近格式,求解一维和二维的分数阶CAblE方程.并用提出的数值方法求解一维和二维情况的两个数值例子,从而说明数值方法的有效性.The cable equation is one of the most fundamental equations for modeling neuronal dynamics.The fractional cable equation has been used to describe nerve fiber.And it has obtained better result in simulating neuron's dynamics behavior.In this paper,we make the first attempt to apply the finite volume method in the solutions of the fractional cable equation in one-dimensional case and two-dimensional case.Firstly the numerical approximation scheme by finite volume method is presented.Then,two numerical examples including one-dimensional case and two-dimensional case are presented by using the numerical technique,which shows the efficiency of the numerical method.国家自然科学基金资助项目(11001090;11026094);福建省自然科学基金资助项目(2010J01011;2010J05009);华侨大学科研启动基金资助项目(08BS507