本文選題：Volterra積分微分方程　切入點：Caputo型分數(shù)次導數(shù)　出處：《上海師范大學》2017年博士論文　論文類型：學位論文

【摘要】：譜方法是求解微分方程的一種重要數(shù)值方法,已被廣泛應用于科學和工程問題的數(shù)值模擬中。譜方法的主要優(yōu)點是計算的高精度,也就是所謂的"無窮階"收斂性,即真解越光滑,譜方法的收斂速度越快。Volterra型積分微分方程和分數(shù)階微分方程等都具有記憶性質,在物理、生物、激光以及人口增長等模型中得到廣泛應用,相關的數(shù)值研究正日益受到重視,并已成為該領域的一個新熱點,而譜方法是一種整體方法,非常適合該類問題的數(shù)值模擬�，F(xiàn)有的針對Volterra型積分、微分方程譜方法的研究主要基于單步格式,并不適合奇性解或長時間的計算。此外,所研究的問題主要是線性的或僅討論光滑解情形,而實際問題大多是非線性的且解呈弱奇異性的。因此本文主要工作之一是研究帶弱奇異核的非線性Volterra型積分微分方程的多步譜方法。我們建立了相關問題的多步譜配置格式,并對所提算法進行了誤差分析,數(shù)值結果表明該方法對光滑解和弱奇性解的模擬都非常有效。對于非線性Caputo型分數(shù)階微分方程的邊值問題,本文將在前人的基礎上提出一種新的譜配置法。為了適應分數(shù)階方程的整體性特點,并克服非線性項的存在所造成的理論分析的困難,我們采用兩種多項式插值,即Legendre-Gauss與Jacobi-Gauss插值,構造相應的Legendre-Jacobi單步譜配置法,并分析了該算法的數(shù)值誤差。數(shù)值算例驗證了該算法的有效性。本文由以下幾個部分組成:在第一章,我們簡單地回顧了譜方法的基本思想及發(fā)展概況,介紹了Volterra積分微分方程與Caputo型分數(shù)階微分方程的問題背景及數(shù)值方法的研究進展。在第二章,我們具體介紹了與本文工作相關的基礎知識:Jacobi多項式及其插值誤差,移位Jacobi多項式,移位Legendre多項式及其插值誤差,并給出了本文工作所需的幾個重要引理。在第三章,對帶有弱奇異核的非線性Volterra積分微分方程提出了一個結構簡單、容易實現(xiàn)的算法,然后詳細地分析了多步譜配置格式的收斂性,獲得了該方法在H1范數(shù)下的hp型誤差估計,最后通過數(shù)值算例展示了該方法的高效性。在第四章,我們考察了 Caputo型分數(shù)階微分方程的兩點邊值問題,提出了基于等價積分方程的Legendre-Jacobi單步譜配置法,詳細地分析了譜配置格式在L2及L∞范數(shù)下的誤差上界,并通過數(shù)值算例驗證了該方法的有效性。最后,對本文工作的主要結果做出總結,分析了其中的不足之處,并在當前工作的基礎上提出改進的方向和措施。
[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，

本文編號：1631103