分数阶偏微分方程的数值解 ——分析和算法 Alternative Title Numerical Solutions of Fractional Partial Differential Equations ---Analyses and Algorithms 张治江 Thesis Advisor 邓伟华 2018-03-20 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 分数阶Laplace方程 回火分数阶导数 Riesz 基 有限元方法 有限差分方法 预处理 分数阶Feynman-Kac 方程 轮廓积分 Abstract 分数阶偏微分方程是一种描述反常扩散现象的行之有效的方法. 基于不同的应用背景, 学者们提出了不同形式的分数阶导数模型. 由于通常情况下分数阶问题不存在可用的解析解, 数值方法在这些模型的实际检验和应用中起着举足轻重的作用. 不同于整数阶导数, 分数阶导数具有(弱) 奇异性、非局部性, 有时甚至还涉及时空耦合性, 这给数值方法的设计、分析和实施带来了不少困难和挑战. 因此, 虽然分数阶偏微分方程的数值解近几十年来受到科研人员的大量关注, 也取得了不少进展, 但它仍然是一个充满活力的学科, 还有不少问题亟待解决. 我们将探讨一些具体模型的数值解决方案. 本文有下述章节构成:第一章, 概述论文的研究背景和主要内容, 包括反常扩散的概念、几种重要的分数阶导数模型、分数阶偏微分方程数值解的研究现状, 以及本文的研究内容和创新之处等.第二章, 探讨回火分数阶 Laplace 方程的 Riesz  基 Galerkin 变分方案. 回火分数阶 Laplace  算子是以积分形式给出的导数, 是分数阶 Laplace  算子的推广. 在统计上它表示对称的回火 \alpha-稳定 Levy  过程的生成子, 其在求解粒子的首次退出时间和逃逸概率方面有着重要的应用. 本章将做三个方面的工作, 即齐次边界条件下回火分数阶 Laplace 方程 Galerkin  弱解的定义及相应变分形式的适定性分析、Riesz  基下的误差估计及有效的数值实施, 和非齐次边界条件的齐次化处理等.第三章, 给出回火分数阶 Laplace 方程的有限差分求解方案, 包括差分格式的构造和相应代数系统的预处理. 我们的差分方案可用于处理一般的非齐次边界条件, 并且它们的收敛速度依赖于精确解在 Ω 上的正则性而不是在全实轴R 上的正则性.第四章, 发展含积分-微分型回火分数阶导数的偏微分方程的变分方案. 该类分数阶模型在文献  [9, 17, 144]  中给出, 它们是相应的 Remiann-Liouville 分数阶模型的推广. 目前求解它们比较完善的数值方法是有限差分法, 其有着严格的稳定性分析和收敛性估计,其它的方法则主要集中于进行一些数值模拟. 本章首先给出回火分数阶导数的一些变分性质, 然后建立一个给定模型的 Galerkin 和 Petrov-Galerkin 有限元求解方案, 并给出严格的理论分析和有效的数值实施.第五章, 探讨欠扩散方程的轮廓积分或有理逼近方案. 轮廓积分和有理逼近方案在求解经典的抛物方程时有着非常广泛的应用, 然而当前利用它们处理时间分数阶偏微分方程的文献却很少. 实际上时间分数阶导数的非局部性质使得这类方法的应用显得更加自然, 也有着更大的优势. 本章首先给出该模型有限元空间半离散格式的收敛性和稳定分析, 然后提供三种轮廓积分或有理逼近方案用于处理时间分数阶导数, 并就每种情况下遇到的具体问题给出了具体的解决方案. 该算法解决了时间分数阶偏微分方程长时间历程计算所面临的计算量和存储量过大的问题.第六章, 介绍回火分数阶 Feynman-Kac 方程的数值逼近方案. 该模型由文献   给出, 被称为向后的回火分数阶 Feynman-Kac 方程, 它的精确解表示粒子轨道的泛函随时间的演化. 处理该模型有两个方面的困难: 一是该方程的时间分数阶导数是时-空耦合的, 二是它反应项前面的符号与我们通常遇到的模型相反. 本章首先给出了对时间分数阶导 数和反应项的合理逼近, 然后在时空耦合范数下证明了全离散格式的稳定性和收敛性.第七章是对本文的总结以及对未来工作的展望. Other Abstract Fractional partial differential equations are an effective method of describing anomalous diffusion phenomena. Based on different application backgrounds, scholars have proposed different forms of fractional derivatives and fractional models. Since there are no available analytical solutions for fractional problems in general, numerical methods play a decisive role in the actual verification and application of these models. Unlike the integer-order derivatives, fractional derivatives have (weak) singularity, non-locality, and sometimes even involve space-time coupling, which bring many difficulties and challenges to the design, analysis, and implementation of numerical methods. Therefore, although the numerical solutions of fractional partial differential equations have received a lot of attention from researchers and have made a lot of progress in recent decades, it is still a dynamic discipline and there are many problems that need to be resolved. In this paper, we will discuss numerical solutions for some specific models. This paper has the following chapters: In Chapter 1, we outline the research backgrounds and main contents of the dissertation, including the concept of anomalous diffusion, several important fractional derivative models, the research status of numerical solution of fractional partial differential equations, and the research contents and innovations of this paper, etc. In Chapter 2, we discuss the Riesz-based Galerkin methods for the tempered fractional Laplacian equation. The tempered fractional Laplacian operator is a derivative given in the form of integral and a generalization of the fractional Laplacian operator. In statistics, it represents the generator of the symmetric tempered  alpha- stable L′evy process, and has an important application in solving the first mean exit time and escape probability of the particles. This chapter will introduce three aspects of work, namely, the definition of Galerkin weak solutions for tempered fractional Laplacian equations and the well-posedness analysis of corresponding variational forms, the error estimation under the Riesz basis and the effective numerical implementation, and the homogeneity of non-homogeneous boundary conditions. In chapter 3, we present the finite difference schemes to solve tempered fractionalLaplacian equation, including the constructions of the difference schemes and the preconditioning of the corresponding discrete systems. Our difference schemes can be used to deal with general nonhomogeneous boundary conditions, and their convergence rates mainly depend on the regularity of the exact solution on Ω rather than the regularity on the whole line R. In Chapter 4, we develop variational schemes for partial differential equations with the integral-differential tempered fractional derivatives. This class of fractional models is given in [9, 17, 144], which is a generalization of the corresponding Remiann-Liouville fractional models. The current numerical methods for solving them are the finite difference methods, which have strict stability analyses and convergence estimations. Other methods are mainly focused on some numerical simulations. In this chapter, we first give some variational properties of the tempered fractional derivatives, and then establish the Galerkin and Petrov-Galerkin finite element schemes for a given model, including the rigorous theoretical analyses and the effective numerical implementations. In Chapter 5, we investigate the contour integral or rational approximation schemes of the subdiffusion equation. Contour integral or rational approximation schemes have a very wide range of applications in solving classical parabolic equations. However, there are few literatures that use them to deal with the time fractional partial differential equations. In fact, the nonlocal property of the time fractional derivative makes the application of this kind of methods appear more natural and has a greater advantage. This chapter first gives the convergence and stability analysis of the model’s finite element spatial semi-discrete scheme, and then provides three contour integral or rational approximation schemes for dealing with the time fractional derivative, and gives solutions to the specific problems encountered in each case. These algorithms solve the problem of excessive calculation and storage capacity for long-time computation of the time fractional partial differential equations. In Chapter 6, numerical approximation schemes of the tempered fractional Feynman-Kac equation are given. This model is presented in  and is called the backward tempered fractional Feynman-Kac equation, which governing the functional distribution first is that the time fractional derivative of the equation is time-space coupled, and the second is that the symbol in front of the response term is contrary to the models we usually encounter. In this chapter, we first give a reasonable approximation for the time fractional derivative and the reaction term. Then we prove the stability and convergence of the fully discrete schemes under the time-space coupled norms. Finally, in Chapter 7, we give the summary of this article and our future work. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225423 Collection 数学与统计学院 Recommended CitationGB/T 7714 张治江. 分数阶偏微分方程的数值解 ——分析和算法[D]. 兰州. 兰州大学,2018.
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