本文選題：分?jǐn)?shù)階延遲微分積分代數(shù)方程　切入點：偏微分代數(shù)方程　出處：《湘潭大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：延遲微分(積分)代數(shù)方程由延遲微分(積分)方程和代數(shù)方程組成,能更好地描述具有記憶性和代數(shù)條件限制的科學(xué)工程問題,如生物學(xué)、自動控制、電磁波、信號處理、系統(tǒng)識別以及多體動力學(xué)等。延遲微分(積分)代數(shù)方程是一類具有時滯性、記憶性、非局部性和代數(shù)約束的微分系統(tǒng),這就給其數(shù)值方法的研究帶來了許多困難。在德國數(shù)學(xué)家G.WLeibniz提出了分?jǐn)?shù)階微積分理論的思想以后,大量實踐表明,分?jǐn)?shù)階微分方程在描述某些實際應(yīng)用問題時比用整數(shù)階微分方程的模型更加精確。近年來,分?jǐn)?shù)階微分(積分)代數(shù)方程也經(jīng)常出現(xiàn)在實際應(yīng)用問題中,越來越受到人們的關(guān)注。這類數(shù)學(xué)模型除具有時滯性、記憶性、非局部性和代數(shù)約束外,其解析解大多含有特殊函數(shù)而不易求得。因此,研究者們提出了幾種求解的迭代算法,如波形松弛法、同倫攝動法、變分迭代法等。其中,變分迭代方法因具有高效、精確、儲存量小等優(yōu)點,已被廣泛的應(yīng)用求解線性和非線性問題。因此,變分迭代法求解延遲積分微分代數(shù)方程和分?jǐn)?shù)階延遲積分微分代數(shù)方程的近似解析解是一種較好的選擇。本文利用變分迭代法求解了幾類延遲微分積分代數(shù)方程。在第一章,闡述了微分(積分)代數(shù)方程以及分?jǐn)?shù)階微分(積分)代數(shù)方程的研究背景以及現(xiàn)狀。在第二章,介紹了變分迭代法。在第三章,針對一類2-指標(biāo)延遲微分積分代數(shù)方程,我們首先利用降指標(biāo)技術(shù),將方程降為1-指標(biāo)的延遲微分積分代數(shù)方程,再根據(jù)方程的特點,選取不同的Lagrange乘子構(gòu)造變分迭代格式求得近似解析解,數(shù)值試驗驗證了方法的收斂性。在第四章,研究了變分迭代法求解延遲偏微分積分代數(shù)方程的收斂性,數(shù)值試驗表明,變分迭代法求解偏微分積分代數(shù)方程能獲得較好的近似解析解。在第五章,應(yīng)用變分迭代法,選取不同的Lagrange乘子,構(gòu)造相應(yīng)的校正泛函,求解了 Caputo分?jǐn)?shù)階延遲微分積分代數(shù)方程,證明了收斂性,數(shù)值試驗說明理論的正確性。最后,對全文總結(jié)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.
【學(xué)位授予單位】：湘潭大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.83

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