解析逼近方法若干问题研究
发布时间：2019-03-12 15:55
【摘要】：现实世界中描述的许多现象都可归结为非线性微分方程,求解非线性微分方程已经成为研究者们面临的关键问题.工程师、物理学家和应用数学家们处理的许多物理问题显示出某些基本特征,而这些基本特征使得相应的问题无法求得精确的解析解.科学技术的发展和符号计算软件的出现促进了非线性微分方程解析逼近方法的发展.同伦分析方法(HAM)和Adomian分解法(ADM)是两种比较常用的解析逼近方法.本文给出这两种方法的一些重要理论改进以及这些改进的非平凡应用.更具体地说,我们完成以下四部分内容.1.提出了一种求解非线性初值问题的混合解析方法,此方法基于同伦分析方法和Laplace变换方法.首先,将一个初值问题转化成初值点在零处的新问题,应用标准同伦分析方法将新的非线性微分方程转换为线性微分方程系统.然后,应用Laplace变换和Laplace逆变换求解所得到的线性初值问题.这些线性初值问题的解析逼近解能够形成给定问题的一个收敛级数解.通过一些非平凡例子证明关于求解高阶变形方程,混合解析逼近方法比标准同伦分析方法更有利.因此,新方法可以应用于求解更复杂的非线性现实问题.2.同伦分析方法的突出特点之一是引入收敛控制参数,收敛控制参数提供了一个简便的方式来调节和控制所得到的级数解的收敛区域和速度.然而,从严格的数学观点来看,收敛控制参数如何实现这个目标?我们得到高阶线性微分方程完整的理论结果.换言之,我们给出由同伦分析方法得到的级数解在某一区间上收敛的严格证明,该区间依赖于收敛控制参数,并且得到该区间上逼近解的绝对误差上界.此外,我们也给出一个确定收敛控制参数有效区域的方法,在所得区域上可以确保级数解收敛.3.基于同伦分析方法求解高阶线性参数边值问题.通过建立所得级数解的显式表达式和大参数与收敛控制参数之间的关系,我们对所求问题的解结构有更深刻的理解.对于大的参数值,通过适当地选择收敛控制参数的值可以得到比较准确的逼近解.与其他解析方法相比,该方法对于求解含大参数的高阶线性边值问题更有效.4.作为经典偏微分方程的推广,分数阶偏微分方程越来越多的被应用于科学的不同领域.与经典偏微分方程相比,分数阶偏微分方程可以更好的模拟现实问题.我们提出一个求解非线性分数阶偏微分方程的新方法.新方法的关键点是在传统Adomian分解法中引入两个参数,称为两参数ADM.已证明两参数ADM逼近解比传统Adomian分解法逼近解更准确.为了说明新方法的适用性和有效性,求解两个非线性分数阶偏微分方程.
[Abstract]:Many phenomena described in the real world can be reduced to nonlinear differential equations. Solving nonlinear differential equations has become a key problem faced by researchers. Many of the physical problems dealt with by engineers physicists and applied mathematicians show some basic characteristics which make it impossible to obtain exact analytical solutions for the corresponding problems. The development of science and technology and the emergence of symbolic computing software promote the development of analytical approximation methods for nonlinear differential equations. Homotopy analysis method (HAM) and Adomian decomposition method (ADM) are two more commonly used analytical approximation methods. In this paper, some important theoretical improvements of these two methods and their nontrivial applications are given. More specifically, we have completed the following four parts. A hybrid analytical method for solving nonlinear initial value problems is proposed. This method is based on homotopy analysis and Laplace transform. Firstly, the initial value problem is transformed into a new problem where the initial point is at zero. The standard homotopy analysis method is used to transform the new nonlinear differential equation into a linear differential equation system. Then, Laplace transform and Laplace inverse transform are used to solve the linear initial value problem. The analytic approximation solutions of these linear initial value problems can form a convergence series solution of a given problem. It is proved by some non-trivial examples that the mixed analytical approximation method is more advantageous than the standard homotopy method in solving higher-order deformation equations. Therefore, the new method can be applied to solve more complex nonlinear reality problems. One of the outstanding characteristics of homotopy analysis is the introduction of convergence control parameters, which provide a simple way to adjust and control the convergence region and speed of the obtained series solutions. However, from a strict mathematical point of view, how can convergence control parameters achieve this goal? We obtain the complete theoretical results of higher order linear differential equations. In other words, we give a strict proof that the series solution obtained by the homotopy analysis method converges on a certain interval, which depends on the convergence control parameters, and obtains the upper bound of the absolute error of the approximation solution on the interval. In addition, we also give a method to determine the effective region of convergence control parameters, on which the convergence of the series solution can be ensured. Based on the homotopy analysis method, the higher order linear parameter boundary value problem is solved. By establishing the explicit expression of the obtained series solution and the relationship between the large parameter and the convergence control parameter, we have a deeper understanding of the solution structure of the problem. For large parameter values, a more accurate approximate solution can be obtained by properly selecting the values of convergence control parameters. Compared with other analytical methods, this method is more effective for solving higher order linear boundary value problems with large parameters. 4. As a generalization of classical partial differential equations, fractional partial differential equations are more and more applied in different fields of science. Compared with classical partial differential equations, fractional partial differential equations can simulate practical problems better. We propose a new method for solving nonlinear fractional partial differential equations. The key point of the new method is to introduce two parameters into the traditional Adomian decomposition method, which is called two-parameter ADM.. It has been proved that the two-parameter ADM approximation solution is more accurate than the traditional Adomian decomposition method. In order to illustrate the applicability and effectiveness of the new method, two nonlinear fractional partial differential equations are solved.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.29
本文编号：2438922
[Abstract]:Many phenomena described in the real world can be reduced to nonlinear differential equations. Solving nonlinear differential equations has become a key problem faced by researchers. Many of the physical problems dealt with by engineers physicists and applied mathematicians show some basic characteristics which make it impossible to obtain exact analytical solutions for the corresponding problems. The development of science and technology and the emergence of symbolic computing software promote the development of analytical approximation methods for nonlinear differential equations. Homotopy analysis method (HAM) and Adomian decomposition method (ADM) are two more commonly used analytical approximation methods. In this paper, some important theoretical improvements of these two methods and their nontrivial applications are given. More specifically, we have completed the following four parts. A hybrid analytical method for solving nonlinear initial value problems is proposed. This method is based on homotopy analysis and Laplace transform. Firstly, the initial value problem is transformed into a new problem where the initial point is at zero. The standard homotopy analysis method is used to transform the new nonlinear differential equation into a linear differential equation system. Then, Laplace transform and Laplace inverse transform are used to solve the linear initial value problem. The analytic approximation solutions of these linear initial value problems can form a convergence series solution of a given problem. It is proved by some non-trivial examples that the mixed analytical approximation method is more advantageous than the standard homotopy method in solving higher-order deformation equations. Therefore, the new method can be applied to solve more complex nonlinear reality problems. One of the outstanding characteristics of homotopy analysis is the introduction of convergence control parameters, which provide a simple way to adjust and control the convergence region and speed of the obtained series solutions. However, from a strict mathematical point of view, how can convergence control parameters achieve this goal? We obtain the complete theoretical results of higher order linear differential equations. In other words, we give a strict proof that the series solution obtained by the homotopy analysis method converges on a certain interval, which depends on the convergence control parameters, and obtains the upper bound of the absolute error of the approximation solution on the interval. In addition, we also give a method to determine the effective region of convergence control parameters, on which the convergence of the series solution can be ensured. Based on the homotopy analysis method, the higher order linear parameter boundary value problem is solved. By establishing the explicit expression of the obtained series solution and the relationship between the large parameter and the convergence control parameter, we have a deeper understanding of the solution structure of the problem. For large parameter values, a more accurate approximate solution can be obtained by properly selecting the values of convergence control parameters. Compared with other analytical methods, this method is more effective for solving higher order linear boundary value problems with large parameters. 4. As a generalization of classical partial differential equations, fractional partial differential equations are more and more applied in different fields of science. Compared with classical partial differential equations, fractional partial differential equations can simulate practical problems better. We propose a new method for solving nonlinear fractional partial differential equations. The key point of the new method is to introduce two parameters into the traditional Adomian decomposition method, which is called two-parameter ADM.. It has been proved that the two-parameter ADM approximation solution is more accurate than the traditional Adomian decomposition method. In order to illustrate the applicability and effectiveness of the new method, two nonlinear fractional partial differential equations are solved.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.29
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