几类分数阶偏微分方程的有限差分方法

发布时间：2019-05-12 14:54

【摘要】：近年来,大量的试验结果表明基于整数阶导数建立的某些模型不能很好地反映现实世界中的一些现象,如反常扩散和复杂粘弹性材料.其中一个主要的原因是传统的整数阶导数由函数的极限定义,其反映的是一个局部的性质.这使得具有非局部特性的分数阶微积分算子受到了广泛关注.然而,由于绝大多数分数阶微分方程的精确解不能被显式给出,对其数值解的研究变得十分必要和重要.鉴于此,本文将引入或改进若干数值方法,以期获得几类空间分数阶和时空分数阶偏微分方程的数值解.在第一章,我们简要回顾了分数阶微积分的发展历程,介绍了求解分数阶微分方程的常用数值方法及其理论,尤其对有限差分方法在这一领域的应用给出了较为详细地介绍,最后给出了将在后续章节频繁用到的一些定义和符号.在第二章,针对一类空间分数阶Schr(o|")dinger波方程,我们导出了它在连续形式下的两个守恒量,提出了一个自封闭的三层线性差分格式,并且讨论了提出格式的守恒能力和精度,借助于数值算例对格式的守恒性能和精度进行了验证.在第三章,考虑了一类强耦合的空间分数阶Schr(o|")dinger方程.本章内容是对第二章工作的推广.同样,我们给出了方程本身具有的两个守恒量,提出了一个非线性的守恒型差分格式,并证明了格式的可解性、稳定性和l∞范数下的收敛性.为提高计算效率,进一步给出了一个线性的守恒型差分格式.数值试验验证了这两种格式的有效性.在第四章,就单个和耦合情形的时空分数阶Schr(o|")dinger方程构造了相应的Crank-Nicolson差分格式及其线性化格式.详细地分析了这些格式的局部截断误差、稳定性和解的存在性,并给出了这两种情形下的数值结果.这项工作的意义在于为这类问题提供了一种新的数值解法,尤其是为耦合问题提供了稳定且有效的线性格式.在第五章,考虑了一类二维半线性的空间分数阶阻尼波方程.该方程有着广泛的应用背景,空间分数阶telegraph方程、sine-Gordon方程和Klein-Gordon方程都可视为它的特殊情形.针对该阻尼波方程,提出了一个有二阶时间精度和四阶空间精度的紧致差分格式,并讨论了格式的可解性和收敛性.为提高计算效率,进一步构造了一个交替方向的紧致差分格式.最后,对应前述三类方程的数值结果验证了格式的有效性.在第六章,提出了一类求解空间分数阶扩散方程的高精度算法.这类算法通过联合分数阶紧致差分逼近和边值方法得到,具有四阶空间精度和四阶、五阶、六阶甚至更高阶的时间精度.为求解产生的大型线性系统,Strang-型、Chan-型和P-型预处理子被引进.当所用边值方法Ak1,k2-稳定时,用GMRES方法求解与Strang-型预处理子相对应的预优系统被证明是快速收敛的.数值算例验证了方法的收敛速率和高精度.在第七章,我们对本文工作做了一个简要的总结,然后罗列了一些有待进一步研究的问题.
[Abstract]:In recent years, a large number of experimental results show that some models based on the integral-order derivative can not well reflect some of the phenomena in the real world, such as abnormal diffusion and complex viscoelastic materials. One of the main reasons is that the traditional integer order derivative is defined by the limit of the function, which reflects a local property. This makes fractional-order micro-integral operators with non-local characteristics widely concerned. However, because the exact solution of most fractional differential equations cannot be given explicit, it is necessary and important to study the numerical solution. In view of this, a number of numerical methods are introduced or modified in order to obtain the numerical solutions of the differential equations of fractional order and space-time fractional order. In the first chapter, we briefly review the development of fractional calculus, and introduce the common numerical method and its theory to solve the fractional differential equation, especially the application of finite difference method in this field. Finally, some definitions and symbols to be used frequently in subsequent chapters are given. In the second chapter, for a class of spatial fractional-order Schr (o | "In this paper, we derive a self-closed three-layer linear difference scheme and verify the conservation and precision of the format by means of a numerical example. In the third chapter, a class of strongly coupled spatial fractional order Schr (o |") dinger equations. The content of this chapter is the promotion of the second chapter. In the same way, we give the two conserved quantities of the equation, and put forward a non-linear conservation type difference scheme, and prove the solvability, stability and the convergence of l-norm. In order to improve the calculation efficiency, a linear conservation type difference scheme is given. The validity of these two formats is verified by numerical experiments. In the fourth chapter, the corresponding Crank-Nicolson difference scheme and its linearized form are constructed for the time-space fractional-order Schr (o | ") dinger equation of a single and a coupling condition. The local truncation error, the stability and the existence of these formats are analyzed in detail. The significance of this work is to provide a new numerical solution for such problems, in particular to provide a stable and effective linear format for the coupling problem. In the fifth chapter, In this paper, a class of two-dimensional and semi-linear space fractional-order damping wave equation is considered. The equation has a wide application background, and the space fractional-order tegph equation, the sine-Gordon equation and the Klein-Gordon equation can be considered as special cases. A compact difference scheme with second-order time precision and fourth-order spatial precision is presented, and the solvability and convergence of the format are discussed. In order to improve the calculation efficiency, a compact difference scheme with alternate directions is further constructed. The validity of the format is verified by the numerical results of the three kinds of equations. In the sixth chapter, a kind of high-precision algorithm for solving the spatial fractional order diffusion equation is proposed. This kind of algorithm is obtained by the joint fractional order compact difference approximation and the edge value method, and has the fourth order spatial precision and the fourth order. Five orders, six orders, and even higher order time precision. In order to solve the large linear system, the Sarang-type, the Chan-type and the P-type pre-processor are introduced. When the edge value methods Ak1 and k2 used are stable, In this paper, the convergence rate and high precision of the method are verified by the GMRES method. The convergence rate and the high precision of the method are verified by numerical examples. In chapter 7, we summarize the work of this paper. A number of issues to be further studied are then listed.
【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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