偏积分微分方程拟小波及紧致差分方法

发布时间：2018-01-13 01:06

本文关键词：偏积分微分方程拟小波及紧致差分方法　出处：《湖南师范大学》2016年博士论文　论文类型：学位论文

【摘要】：随着科学技术的发展,人类发现了一类重要的方程：积分微分方程.这类方程出现在很多领域,如：具有记忆材料中的热传导、粘弹性力学、人口动力学等问题.由于这类方程的积分微分项通常带有弱奇异性,因此这类方程的求解是一件困难的事情.而解决这类积分微分方程也是当今研究的热门课题之一本文主要采用拟小波方法、紧差分方法、有限差分方法来求解三类不同的积分微分方程.第一类方程是无界区域上的带正定记忆项的积分微分方程.第二类和第三类方程均为带弱奇异项的积分微分方程.本文由七章组成.第一章主要介绍积分微分方程的历史背景、研究动态,以及本论文的研究内容的结构.第二章和第三章主要采用拟小波方法研究第一类积分微分方程,第四章主要采用紧致差分方法研究第二类方程,第五章主要采用交替方向隐式欧拉方法及有限差分方法研究第三类方程,第六章主要采用交替方向隐式Crank-Nicolson方法及有限差分方法研究第三类方程.第七章为总结与展望.在第二章对第一类方程的研究中,我们首次采用了拟小波方法求解一类无界区域上的积分微分方程.首先在无界区间上任意选取一定点x0,再取定任意正常数p,这样我们就可以得到无界区域上的一有限区域[x0-p,x0+p]这样我们就可以在这段区域上求出方程的数值解.一方面,由于该有限区域是任意确定的,而且拟小波是局部可解的,因此在端点附近的值与准确值之间存在较大的误差,为了得到更准确的数值解,我们在求解误差时将舍去一些误差较大的数值解,这在后文中将详细描述.另一方面,由于数x0，p的任意性,我们可以模拟出无界区域上任意区间的数值解.在对方程进行数值求解时,我们在时间上采用向前欧拉方法离散,空间方向采用拟小波方法离散,对于积分项采用三角形法则来逼近,即取右端点上的积分值作为逼近值.在这章中我们不仅求解了一维模型,还研究了二维模型,并给出了几个数值例子加以证明.第三章我们采用Crank-Nicolson方法结合拟小波方法对第二章同类无界区域上的积分微分方程进行了研究探讨.我们同样采用上述方法来处理无界区间.时间方向采用Crank_Nicolson方法,空间方向同样用拟小波方法,而积分项则采用连续分片线性插值逼近方法来逼近.在这一章里我们同样研究了一维和二维模型,并采用了第二章的数值例子进行比较.不管是在工程还是在数学领域,有限差分法被大部分工程师和研究学者采纳用来解决各类问题.这是因为差分法是一种简单有效的办法,所以被大部分人采用.同样过去许多学者采用差分法研究了积分微分方程,然而采用紧差分方法求解弱奇异积分微分方程却工作较少.由于紧差分是一种四阶格式,而有限差分方法是一种二阶格式,因此本文第四章主要采用紧差分方法代替有限差分方法求解带弱奇异项的积分微分方程.在得到离散格式后,我们还严格证明了全离散格式的稳定性和收敛性分析.并用数值例子证明了分析的准确性.在了解到弱奇异积分微分方程在时间上总是达不到丰满阶之后,我们对一种新的区间剖分方法产生兴趣：等级网格剖分方法.这是因为采用等级网格方法剖分区间,使得区间在奇异点附近比较密集,在远离奇异点附近比较稀疏,这样有效的弥补了方程的解的奇异性.另一方面,在了解到求解二维问题需要大量的存储空间来存储过去的数据之后,我们想到了用交替方向隐式有限差分法来缓解存储问题,减少程序运行时间.因此,在第五章采用基于非一致网格的交替方向有限差分法来研究弱奇异问题,在时间方向采用隐式欧拉格式,空间方向采用二阶差分格式离散,得到收敛阶为O(k+h2x+h2y)还证明了该方法的稳定性和收敛性.最后我们给出了两个数值例子加以证明.第六章是在第五章的基础上对等级网格法处理弱奇异核问题的进一步研究.第六章我们进一步研究了二阶Crank-Nicolson格式来处理弱奇异核问题：时间方向采用交替方向隐式Crank-Nicolson格式替换隐式欧拉格式离散,空间方向同样采用二阶差分格式离散,这样得到的离散格式具有收敛阶O(k2+h2x+h2y)在这章同样给出了该离散格式的稳定性和收敛性分析,并采用第五章的两个例子证明理论分析的可靠性.
[Abstract]:With the development of science and technology, human has found an important class of equations: Integro differential equations. These equations appear in many areas, such as: heat conduction in the material with memory, viscoelasticity, population dynamics and so on. Because of these integral differential equation usually with weak singularity, thus solving the equation is difficult. To solve this class of Integro differential equations is the current hot topic of research. This paper mainly adopts quasi wavelet method, compact difference method, finite difference method to solve the three kinds of integral differential equations. The first equation is positive definite integral differential equation of unbounded memory the band. Class second and third kinds of equations are Integro differential equations with weakly singular term. This paper consists of seven chapters. The first chapter mainly introduces the historical background, Integro differential equations and the dynamic research. The structure of the research content. The second chapter and the third chapter mainly adopts quasi wavelet method of first class of Integro differential equations, the fourth chapter mainly uses the compact difference method of second kinds of equations, the fifth chapter mainly adopts alternating direction implicit Euler method and the finite difference method of third kinds of equations, the sixth chapter mainly adopts the alternating direction implicit the Crank-Nicolson method and the finite difference method of third kinds of equations. The seventh chapter is the summary and outlook. In the second chapter of the first equation, we first use the quasi wavelet method for solving a class of Integro differential equations on unbounded domains. First in the unbounded interval arbitrary point x0, then any normal number P, so we can get a limited area of [x0-p on unbounded domain x0+p], so that we can in this area for the numerical solution of the equation. On the one hand, due to the The limited area is arbitrarily determined, and the quasi wavelet is locally solvable, so near the end of the value and the accurate value between the larger error, in order to get a more accurate numerical solution, we will give some numerical error in solving the solution error, which will be described in detail in this paper after another. Hand, as the number of x0, any p, we can simulate the numerical solution of arbitrary interval on unbounded domains. In the numerical solution of the equation, we use the forward Euler method in time discretization, spatial direction by using the quasi wavelet method for discrete integral using triangle rule approximation, namely takes the integral right the endpoint value as the approximate value. In this chapter, we not only solve the one-dimensional model, also studied the two-dimensional model, and several numerical examples are given to prove it. In the third chapter, we use the Crank-Nicolson method combined with quasi wavelet. The study of integral differential equations on unbounded domains in chapter second similar method. We also used the method to deal with unbounded intervals. The direction of time by using Crank_Nicolson method, spatial direction with the same quasi wavelet method, and integral using continuous piecewise linear interpolation approximation method to approximate. In this chapter we also study on a two-dimensional model, and using numerical examples. The second chapter analyzed. Whether in engineering or in the field of mathematics, the finite difference method is adopted to most engineers and researchers to solve all kinds of problems. This is because the difference method is a simple and effective way, so most people are using. The same past many scholars by difference method of integral differential equation to study, but the compact finite difference method for weakly singular integral differential equation has less work. Because of the compact difference is A four order scheme, and the finite difference method is a kind of two order format, so the fourth chapter of this paper mainly adopts the compact difference method instead of the finite difference method for solving Integro differential equations with weakly singular terms. In the discrete format, we strictly prove the stability and convergence of the fully discrete scheme. The accuracy of the analysis is proved by a numerical example. After understanding to weakly singular Integro differential equation in time always reach the fullness of our order, a new interval subdivision method: interest level mesh subdivision. This is because the level of region subdivision grid method, makes the interval in the singular point near the more intensive, relatively sparse in the vicinity of far away from the singular points, thus effectively compensate for equations singularity. On the other hand, to solve the two-dimensional problem in understanding the needs of large storage space to store the data in the past After that, we thought of using the alternating direction implicit finite difference method to solve the storage problem, reduce the running time of the program. Therefore, to study the problem of using weakly singular non uniform grid alternating direction based on finite difference method in the fifth chapter, using the implicit Euler scheme in time, space by two order difference scheme get the order of convergence is discrete, O (k+h2x+h2y) also proved the stability and convergence of the method. Finally, we give two numerical examples to prove it. The sixth chapter is on the basis of the fifth chapter with further study the problem of weakly singular kernel level grid method. In the sixth chapter, we further study the two order Crank-Nicolson format to deal with weakly singular kernel problem: the time direction by alternating direction implicit Crank-Nicolson scheme to replace the implicit Euler scheme, spatial direction using the same two order difference scheme, this kind of The discrete scheme has convergence order O (k2+h2x+h2y). In this chapter, the stability and convergence analysis of the discrete scheme are also given. Two examples of the fifth chapter are used to prove the reliability of the theoretical analysis.

【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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