非线性Volterra积分微分方程及分数阶微分方程的谱配置法

发布时间：2018-03-18 20:06

  本文选题：Volterra积分微分方程　切入点：Caputo型分数次导数　出处：《上海师范大学》2017年博士论文　论文类型：学位论文

【摘要】：谱方法是求解微分方程的一种重要数值方法,已被广泛应用于科学和工程问题的数值模拟中。谱方法的主要优点是计算的高精度,也就是所谓的"无穷阶"收敛性,即真解越光滑,谱方法的收敛速度越快。Volterra型积分微分方程和分数阶微分方程等都具有记忆性质,在物理、生物、激光以及人口增长等模型中得到广泛应用,相关的数值研究正日益受到重视,并已成为该领域的一个新热点,而谱方法是一种整体方法,非常适合该类问题的数值模拟。现有的针对Volterra型积分、微分方程谱方法的研究主要基于单步格式,并不适合奇性解或长时间的计算。此外,所研究的问题主要是线性的或仅讨论光滑解情形,而实际问题大多是非线性的且解呈弱奇异性的。因此本文主要工作之一是研究带弱奇异核的非线性Volterra型积分微分方程的多步谱方法。我们建立了相关问题的多步谱配置格式,并对所提算法进行了误差分析,数值结果表明该方法对光滑解和弱奇性解的模拟都非常有效。对于非线性Caputo型分数阶微分方程的边值问题,本文将在前人的基础上提出一种新的谱配置法。为了适应分数阶方程的整体性特点,并克服非线性项的存在所造成的理论分析的困难,我们采用两种多项式插值,即Legendre-Gauss与Jacobi-Gauss插值,构造相应的Legendre-Jacobi单步谱配置法,并分析了该算法的数值误差。数值算例验证了该算法的有效性。本文由以下几个部分组成:在第一章,我们简单地回顾了谱方法的基本思想及发展概况,介绍了Volterra积分微分方程与Caputo型分数阶微分方程的问题背景及数值方法的研究进展。在第二章,我们具体介绍了与本文工作相关的基础知识:Jacobi多项式及其插值误差,移位Jacobi多项式,移位Legendre多项式及其插值误差,并给出了本文工作所需的几个重要引理。在第三章,对带有弱奇异核的非线性Volterra积分微分方程提出了一个结构简单、容易实现的算法,然后详细地分析了多步谱配置格式的收敛性,获得了该方法在H1范数下的hp型误差估计,最后通过数值算例展示了该方法的高效性。在第四章,我们考察了 Caputo型分数阶微分方程的两点边值问题,提出了基于等价积分方程的Legendre-Jacobi单步谱配置法,详细地分析了谱配置格式在L2及L∞范数下的误差上界,并通过数值算例验证了该方法的有效性。最后,对本文工作的主要结果做出总结,分析了其中的不足之处,并在当前工作的基础上提出改进的方向和措施。
[Abstract]:Spectral method is an important numerical method for solving differential equations, which has been widely used in numerical simulation of scientific and engineering problems. The main advantage of spectral method is the high accuracy of calculation, that is, the so-called "infinite order" convergence. That is, the more smooth the true solution, the faster the convergence of spectral methods. Volterra type integro-differential equations and fractional differential equations have memory properties, which are widely used in physical, biological, laser and population growth models. The related numerical research has been paid more and more attention, and has become a new hotspot in this field. The spectral method is a global method, which is very suitable for the numerical simulation of this kind of problems. The spectral methods of differential equations are mainly based on single-step schemes and are not suitable for singularities or long-time calculations. In addition, the problems studied are mainly linear or only for smooth solutions. However, the practical problems are mostly nonlinear and the solutions are weakly singular. Therefore, one of the main work of this paper is to study the multistep spectral method for nonlinear Volterra type integro-differential equations with weakly singular kernels. The numerical results show that the proposed method is very effective for the simulation of smooth solutions and weak singularities. For the boundary value problems of nonlinear fractional differential equations of Caputo type, the numerical results show that the proposed method is very effective for the simulation of smooth solutions and weak singularities. In this paper, a new spectral collocation method is proposed on the basis of predecessors. In order to adapt to the integral characteristics of fractional order equations and overcome the difficulty of theoretical analysis caused by the existence of nonlinear terms, we adopt two kinds of polynomial interpolation. That is, Legendre-Gauss and Jacobi-Gauss interpolation, construct the corresponding Legendre-Jacobi single-step spectrum collocation method, and analyze the numerical error of the algorithm. Numerical examples verify the effectiveness of the algorithm. This paper is composed of the following parts: in the first chapter, We briefly review the basic ideas and development of spectral methods, introduce the background of Volterra integrodifferential equations and fractional differential equations of Caputo type and the research progress of numerical methods. In this paper, we introduce the basic knowledge related to the work of this paper: Jacobi polynomials and their interpolation errors, shift Jacobi polynomials, shift Legendre polynomials and their interpolation errors, and give some important lemmas for the work of this paper. A simple and easy algorithm for nonlinear Volterra integro-differential equations with weakly singular kernels is presented. Then the convergence of multistep spectral collocation scheme is analyzed in detail, and the hp-type error estimates of this method under H1-norm are obtained. In chapter 4th, we investigate the two-point boundary value problem of fractional differential equation of Caputo type, and propose a Legendre-Jacobi single-step spectrum collocation method based on equivalent integral equation. The error upper bound of spectral collocation scheme under L _ 2 and L _ 鈭，

本文编号：1631103