分段连续型延迟微分方程的变分迭代法

发布时间：2018-03-22 14:51

本文选题：变分迭代法　切入点：Lagrange乘子　出处：《广东工业大学》2017年硕士论文　论文类型：学位论文

【摘要】：变分迭代法是求解线性和非线性微分方程的一种非常有效的方法,通过变分迭代法可以获得这些方程的近似解析解或精确解.在求解的过程中无需将方程的非线性部分进行任何线性化、离散化或者引入摄动参数,从而减少计算量.目前,它在求解振荡方程、波方程、延迟微分方程及分数阶微分方程等非线性问题中被广泛应用.本文主要将变分迭代法用于求解分段连续型延迟微分方程的初值问题.本文的内容和结果如下:第一章介绍了变分迭代法的研究背景,概述了近年来该方法的国内外研究现状,并给出了本文的研究内容.第二章主要介绍求解线性分段连续型延迟微分方程初值问题的变分迭代法.首先用该方法构造出了迭代格式,代入迭代初值后获得一个迭代解序列,然后从理论上证明了该迭代解序列收敛于问题的精确解,最后通过数值实验加以验证.第三章主要介绍求解非线性分段连续型延迟微分方程初值问题的变分迭代法.首先用变分迭代方法的基础理论求出了拉格朗日乘子,然后选取迭代初值,最后代入迭代格式中获得一个迭代解序列,并从理论上证明了该迭代解序列是收敛的,用实例表明该结果是正确的.第四章主要介绍求解分段连续型延迟偏微分方程初值问题的变分迭代法.首先确定出了拉格朗日乘子,然后将拉格朗日乘子和选取的初始值代入迭代格式中得到了一个迭代解序列,最后证明了该迭代解序列的收敛性,并通过具体例子验证了该结论的正确性.
[Abstract]:Variational iterative method is a very effective method for solving linear and nonlinear differential equations. The approximate analytical solutions or exact solutions of these equations can be obtained by variational iterative method. In the process of solving these equations, there is no need to linearize, discretize or introduce perturbation parameters to the nonlinear part of the equation, thus reducing the computational complexity. It's solving the oscillation equation, the wave equation, Delay differential equations and fractional differential equations are widely used in nonlinear problems. In this paper, variational iterative method is mainly used to solve the initial value problems of piecewise continuous delay differential equations. The contents and results of this paper are as follows:. In the first chapter, the background of variational iterative method is introduced. The research status of this method at home and abroad in recent years is summarized. In the second chapter, the variational iterative method for solving the initial value problem of linear piecewise continuous delay differential equations is introduced. Then it is proved theoretically that the iterative solution sequence converges to the exact solution of the problem. In chapter 3, the variational iterative method for solving the initial value problem of nonlinear piecewise continuous delay differential equations is introduced. The Lagrangian multiplier is obtained by using the basic theory of variational iterative method. Then the initial value of the iteration is selected and an iterative solution sequence is obtained in the iterative scheme. It is proved theoretically that the iterative solution sequence is convergent. In chapter 4, the variational iterative method for solving the initial value problem of piecewise continuous delay partial differential equations is introduced. Then the Lagrange multiplier and the selected initial value are substituted into the iterative scheme to obtain an iterative solution sequence. Finally, the convergence of the iterative solution sequence is proved, and the correctness of the conclusion is verified by an example.
【学位授予单位】：广东工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

【参考文献】