延迟微分方程和积分代数方程的谱方法研究

发布时间：2018-06-20 09:13

本文选题：Volterra延迟积分微分方程 + 积分代数方程　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：谱方法、有限元法、有限差分法都是求解线性与非线性微分方程的有效数值方法。谱方法是一类对微分方程空间变量离散的方法,它主要由试探函数(也称基函数或展开函数)和检验函数组成。谱方法中的试探函数为无穷可微的整体函数(通常是奇异或非奇异Sturm-Liouville问题的特征函数)。根据检验函数的选取不同,可将谱方法分为谱Galerkin方法、谱配置法(也称拟谱方法)和谱Tau方法。谱方法最大优势在于它具有所谓的“无穷阶收敛性”,但这必须要求原问题的真解能够达到充分光滑,这样就导致了谱方法的缺点是不能灵活地适应复杂区域的计算。本文将谱方法中的谱配置法和谱Tau方法引入到延迟微分方程与积分代数方程上来,并对收敛性进行了深入研究。谱配置法是将检验函数取为以配置点为中心的Dirac-δ函数,这样使得微分方程在配置点上精确成立。将选取Legendre-Gauss型求积公式节点为配置点、选取Legendre多项式为试探基函数的配置法称为Legendre-Gauss配置法。本文将运用Legendre-Gauss配置法数值求解非线性中立型延迟微分方程和非线性Volterra型延迟积分微分方程。这两类方程的解在求解区间内的整体光滑性并不理想,这是因为真解在求解区间内个别点上的光滑性很差,从而导致整体光滑性不佳,而这些点是由方程中延迟函数θ(t)所确定的。为解决这一难题,本文提出了多区域Legendre-Gauss配置法。该方法是将求解区间进行充分剖分,从而保证方程真解在每个子区间都能够充分光滑;然后分别在每个子区间内求其配置解,进而获得全局数值解。按此方法获得的数值解是能够具备谱精度的,即真解只要能在由θ(t)确定的那些点之外充分光滑,则数值解就能够做到“无穷阶收敛”。与以往的谱Tau方法不同,Lanczos和Ortiz提出了一种便于操作的Lanczos Tau方法。这种方法不需要进行积分近似,它是将微分方程直接近似转化为代数方程组。本文运用Lanczos Tau方法来求解比例型线性Volterra延迟积分微分方程。由数值算例的对比结果可见,相比Legendre配置法的高精度,Lanczos Tau方法的优势在于它的高效性。在达到相同收敛阶时,Lanczos Tau方法所用时间要远远少于Legendre配置法所使用的。本文同时给出了Lanczos Tau方法在一般情形下的收敛性分析,并指出了决定其收敛速度的关键因素,这些理论结果在以往的研究成果中是比较少见的。配置法中,取等距节点jh(j=0,±1,±2,···,h0)映射到求解区域的对应点为配置点,取Sinc基函数为试探基函数的配置法称为Sinc配置法。Sinc配置法是另一种高精度的数值方法,它不需要方程具有较高的正则性。作为试探基函数的Sinc基函数能够对奇异、振荡等问题给出很好的逼近,并同时具备良好的稳定性,这使得Sinc配置法在处理复杂方程时具有许多优势。本文运用Sinc配置法求解比例型线性Volterra延迟积分微分方程和具有指标1的积分代数方程,这是Sinc配置法应用的新尝试。通过误差分析可知,Sinc配置法是能够以指数阶收敛的高精度数值方法。
[Abstract]:The spectral method, finite element method and finite difference method are all effective numerical methods for solving linear and nonlinear differential equations. The spectral method is a kind of method to discrete the spatial variables of differential equations. It is composed mainly of exploratory functions (also known as basic functions or expansion functions) and test functions. The test function in the spectral method is a infinitely differentiable function ( The spectral method can be divided into spectral Galerkin method, spectral collocation method (also called pseudo spectral method) and spectral Tau method according to the selection of the test function. The greatest advantage of the spectral method is that it has the so-called "infinite order convergence", but it must require the true solution of the original problem. The spectral method and the spectral Tau method are introduced to the delay differential equation and the integral algebraic equation, and the convergence is deeply studied. The spectral collocation method is to take the test function as the point in the configuration point. The Dirac- delta function of the heart makes the differential equation set up accurately at the point of configuration. The Legendre-Gauss type formula node is selected as the configuration point, and the configuration method of selecting the Legendre polynomial as the basis function is called the Legendre-Gauss configuration method. This paper will use the Legendre-Gauss configuration method to solve the nonlinear neutral delay differential equation. And the nonlinear Volterra type delay integral differential equation. The solution of the two kinds of equations is not ideal in the solution interval. This is because the smoothness of the true solution on the individual points in the solution interval is very poor, which leads to the poor overall smoothness, and these points are determined by the delay function theta (T) in the equation. In this paper, a multi region Legendre-Gauss configuration method is proposed. This method is to fully dissection the solution interval, so that the true solution of the equation can be fully smooth in each subinterval, and then the solution is obtained in each subinterval, and then the global numerical solution is obtained. As long as the solution can be fully smooth beyond those determined by theta (T), then the numerical solution can achieve "infinite order convergence". Unlike the previous spectral Tau method, Lanczos and Ortiz proposed a Lanczos Tau method which is easy to operate. This method does not require integral approximation. It is a direct approximation of the differential equation into an algebraic equation. In this paper, the Lanczos Tau method is used to solve the proportional linear Volterra delay integral differential equation. The comparison between the numerical examples shows that the advantage of the Lanczos Tau method is its high efficiency compared with the high precision of the Legendre configuration method. The time of the Lanczos Tau method is far less than the Legendre configuration when the same convergence order is reached. At the same time, this paper gives the convergence analysis of the Lanczos Tau method in the general case, and points out the key factors to determine its convergence speed. These theoretical results are rare in the previous research results. In the configuration method, the equivalent node JH (j=0, + 1, + 2, and H0) is mapped to the corresponding point of the solution area. Point, the configuration method of taking the Sinc basis function as the basis function is called the Sinc configuration method.Sinc configuration method is another high precision numerical method, it does not need the high regularity of the equation. The Sinc basis function of the test basis function can give good compel to the singularity, oscillation and so on, and also has good stability, which makes Sin C configuration method has many advantages in dealing with complex equations. This paper uses Sinc configuration method to solve proportional linear Volterra delay integral differential equation and integral algebraic equation with index 1. This is a new attempt for the application of Sinc configuration method. By error analysis, it is known that Sinc configuration method is a high-precision numerical method that can converge to exponential order.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

【相似文献】