本文選題：時(shí)滯拋物型方程 + 緊差分格式��； 參考：《延邊大學(xué)》2017年碩士論文

【摘要】：在自然界中,時(shí)滯現(xiàn)象普遍存在且無(wú)法避免,這也是影響系統(tǒng)穩(wěn)定性及其性能的主要原因之一,時(shí)滯微分方程在理學(xué)、工學(xué)等眾多領(lǐng)域中都有著廣泛應(yīng)用.過(guò)去,人們?cè)谘芯刻祗w力學(xué)、物理學(xué)、動(dòng)力系統(tǒng)等學(xué)科中的問(wèn)題時(shí),總認(rèn)為所考慮的系統(tǒng)服從這樣一個(gè)規(guī)律,即系統(tǒng)將來(lái)的狀態(tài)僅由系統(tǒng)當(dāng)前的狀態(tài)決定并用相應(yīng)的模型加以刻畫(huà).然而,隨著人們對(duì)許多自然現(xiàn)象有了更深入的分析后發(fā)現(xiàn),現(xiàn)實(shí)世界中,系統(tǒng)的狀態(tài)除了依賴當(dāng)前發(fā)展?fàn)顟B(tài)也依賴過(guò)去的發(fā)展系統(tǒng).在多數(shù)情況下,若用忽略時(shí)滯的方法來(lái)降低問(wèn)題的難度,會(huì)給系統(tǒng)帶來(lái)比較大的負(fù)面影響,但也正因?yàn)橛袝r(shí)滯項(xiàng),其理論的分析難度較大,想獲得其精確解的解析表達(dá)式是很困難的.所以,我們?cè)诮鉀Q實(shí)際問(wèn)題的時(shí)候,時(shí)滯微分方程精確解的得出一般都用其數(shù)值解來(lái)替代.這一研究彌補(bǔ)了理論上的不足,同時(shí)具有重要的現(xiàn)實(shí)意義.本文闡述了如何構(gòu)造時(shí)滯拋物型方程的緊差分格式,同時(shí)也介紹了其對(duì)應(yīng)的數(shù)值格式理論分析.第一章主要講述了專家學(xué)者們對(duì)有關(guān)時(shí)滯微分方程的數(shù)值方法研究的多年進(jìn)展?fàn)顩r,以及有關(guān)時(shí)滯微分方程研究的背景和意義,并且說(shuō)明了本文的主要研究?jī)?nèi)容及意義.第二章主要用了差分離散的方法為一維非線性時(shí)滯拋物型方程的初邊值問(wèn)題構(gòu)造出一個(gè)緊差分格式,同時(shí)用能量分析法證明了其在該格式下解的存在唯一性、無(wú)條件穩(wěn)定性和在L∞范數(shù)下階數(shù)為O(T2+ 4 的收斂性.最后,用一個(gè)數(shù)值算例說(shuō)明該格式具有可行性.第三章闡述了如何構(gòu)造二維時(shí)滯拋物型方程初邊值問(wèn)題的緊差分格式,這里,我們用交替方向的技巧來(lái)提高計(jì)算效率,并對(duì)緊差分格式進(jìn)行求解,接著研究了解的先驗(yàn)估計(jì)式和穩(wěn)定性.最后,用一個(gè)數(shù)值算例說(shuō)明該格式具有可行性.
[Abstract]:In nature, the phenomenon of delay exists widely and cannot be avoided, which is one of the main reasons that affect the stability and performance of systems. Delay differential equations are widely used in many fields, such as science, engineering and so on. In the past, when people studied the problems in astromechanics, physics, dynamical systems, and other subjects, they always thought that the system under consideration was based on such a rule. That is, the future state of the system is only determined by the current state of the system and described by the corresponding model. However, with more in-depth analysis of many natural phenomena, it is found that in the real world, the state of the system depends not only on the current state of development, but also on the development system of the past. In most cases, if we use the method of neglecting time delay to reduce the difficulty of the problem, it will bring more negative effects to the system, but it is also difficult to analyze the theory because of the delay term. It is difficult to obtain an analytical expression of its exact solution. Therefore, when we solve practical problems, the exact solutions of delay differential equations are generally replaced by their numerical solutions. This research makes up for the deficiency in theory and has important practical significance at the same time. This paper describes how to construct a compact difference scheme for the parabolic equation with time delay, and also introduces its corresponding numerical scheme theory analysis. In the first chapter, the progress of the numerical methods of delay differential equations, the background and significance of the research on delay differential equations are described, and the main contents and significance of this paper are explained. In the second chapter, a compact difference scheme is constructed for the initial boundary value problem of one-dimensional nonlinear parabolic equations with delay by using the method of difference discretization, and the existence and uniqueness of its solution under the scheme are proved by energy analysis. Unconditional stability and convergence of order O(T2 4 under L 鈭，

本文編號(hào)：1840023