几类延迟微分积分代数方程的变分迭代法

发布时间：2018-03-22 19:44

本文选题：分数阶延迟微分积分代数方程　切入点：偏微分代数方程　出处：《湘潭大学》2017年硕士论文　论文类型：学位论文

【摘要】：延迟微分(积分)代数方程由延迟微分(积分)方程和代数方程组成,能更好地描述具有记忆性和代数条件限制的科学工程问题,如生物学、自动控制、电磁波、信号处理、系统识别以及多体动力学等。延迟微分(积分)代数方程是一类具有时滞性、记忆性、非局部性和代数约束的微分系统,这就给其数值方法的研究带来了许多困难。在德国数学家G.WLeibniz提出了分数阶微积分理论的思想以后,大量实践表明,分数阶微分方程在描述某些实际应用问题时比用整数阶微分方程的模型更加精确。近年来,分数阶微分(积分)代数方程也经常出现在实际应用问题中,越来越受到人们的关注。这类数学模型除具有时滞性、记忆性、非局部性和代数约束外,其解析解大多含有特殊函数而不易求得。因此,研究者们提出了几种求解的迭代算法,如波形松弛法、同伦摄动法、变分迭代法等。其中,变分迭代方法因具有高效、精确、储存量小等优点,已被广泛的应用求解线性和非线性问题。因此,变分迭代法求解延迟积分微分代数方程和分数阶延迟积分微分代数方程的近似解析解是一种较好的选择。本文利用变分迭代法求解了几类延迟微分积分代数方程。在第一章,阐述了微分(积分)代数方程以及分数阶微分(积分)代数方程的研究背景以及现状。在第二章,介绍了变分迭代法。在第三章,针对一类2-指标延迟微分积分代数方程,我们首先利用降指标技术,将方程降为1-指标的延迟微分积分代数方程,再根据方程的特点,选取不同的Lagrange乘子构造变分迭代格式求得近似解析解,数值试验验证了方法的收敛性。在第四章,研究了变分迭代法求解延迟偏微分积分代数方程的收敛性,数值试验表明,变分迭代法求解偏微分积分代数方程能获得较好的近似解析解。在第五章,应用变分迭代法,选取不同的Lagrange乘子,构造相应的校正泛函,求解了 Caputo分数阶延迟微分积分代数方程,证明了收敛性,数值试验说明理论的正确性。最后,对全文总结并做展望。
[Abstract]:The delay differential (integral) algebraic equation is composed of the delay differential (integral) equation and the algebraic equations, which can better describe the scientific engineering problems with memory and algebraic conditions, such as biology, automatic control, electromagnetic wave, signal processing, etc. Delay differential (integral) algebraic equations are a class of differential systems with delay, memory, nonlocality and algebraic constraints. After the German mathematician G.WLeibniz put forward the idea of fractional calculus theory, a great deal of practice shows that, Fractional differential equations are more accurate in describing some practical problems than models of integer order differential equations. In recent years, fractional differential (integral) algebraic equations often appear in practical applications. More and more attention has been paid to this kind of mathematical models. Except for their delay, memory, nonlocality and algebraic constraints, the analytical solutions of these mathematical models are difficult to obtain because of their special functions. Therefore, some iterative algorithms for solving these mathematical models are proposed. For example, wave relaxation method, homotopy perturbation method, variational iteration method and so on. The variational iteration method has been widely used to solve linear and nonlinear problems because of its advantages of high efficiency, accuracy and small storage. The variational iterative method is a good choice for solving delay integro-differential algebraic equations and fractional delay integro-differential algebraic equations. In the first chapter, the variational iterative method is used to solve several kinds of delay integro-differential algebraic equations. The research background and present situation of differential (integral) algebraic equations and fractional differential (integral) algebraic equations are described. In chapter 2, variational iterative method is introduced. In chapter 3, for a class of 2-index delay differential integral algebraic equations, In this paper, we first reduce the equation to 1-index delay differential integral algebraic equation by using the reduced index technique. Then, according to the characteristics of the equation, we select different Lagrange multipliers to construct the variational iterative scheme to obtain the approximate analytical solution. In chapter 4, the convergence of variational iterative method for solving delayed partial differential integral algebraic equations is studied. The variational iterative method can obtain better approximate analytical solutions for partial differential integral algebraic equations. In Chapter 5, using the variational iteration method, different Lagrange multipliers are selected to construct the corresponding correction functional. In this paper, the Caputo fractional delay differential integral algebraic equation is solved, the convergence is proved, and the correctness of the theory is proved by numerical experiments. Finally, the paper is summarized and prospected.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83

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