發(fā)布時間：2018-05-03 20:45

  本文選題：時滯拋物型方程 + 緊差分格式�。� 參考：《延邊大學》2017年碩士論文

【摘要】：在自然界中,時滯現(xiàn)象普遍存在且無法避免,這也是影響系統(tǒng)穩(wěn)定性及其性能的主要原因之一,時滯微分方程在理學、工學等眾多領域中都有著廣泛應用.過去,人們在研究天體力學、物理學、動力系統(tǒng)等學科中的問題時,總認為所考慮的系統(tǒng)服從這樣一個規(guī)律,即系統(tǒng)將來的狀態(tài)僅由系統(tǒng)當前的狀態(tài)決定并用相應的模型加以刻畫.然而,隨著人們對許多自然現(xiàn)象有了更深入的分析后發(fā)現(xiàn),現(xiàn)實世界中,系統(tǒng)的狀態(tài)除了依賴當前發(fā)展狀態(tài)也依賴過去的發(fā)展系統(tǒng).在多數(shù)情況下,若用忽略時滯的方法來降低問題的難度,會給系統(tǒng)帶來比較大的負面影響,但也正因為有時滯項,其理論的分析難度較大,想獲得其精確解的解析表達式是很困難的.所以,我們在解決實際問題的時候,時滯微分方程精確解的得出一般都用其數(shù)值解來替代.這一研究彌補了理論上的不足,同時具有重要的現(xiàn)實意義.本文闡述了如何構(gòu)造時滯拋物型方程的緊差分格式,同時也介紹了其對應的數(shù)值格式理論分析.第一章主要講述了專家學者們對有關(guān)時滯微分方程的數(shù)值方法研究的多年進展狀況,以及有關(guān)時滯微分方程研究的背景和意義,并且說明了本文的主要研究內(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滯拋物型方程的初邊值問題構(gòu)造出一個緊差分格式,同時用能量分析法證明了其在該格式下解的存在唯一性、無條件穩(wěn)定性和在L∞范數(shù)下階數(shù)為O(T2+ 4 的收斂性.最后,用一個數(shù)值算例說明該格式具有可行性.第三章闡述了如何構(gòu)造二維時滯拋物型方程初邊值問題的緊差分格式,這里,我們用交替方向的技巧來提高計算效率,并對緊差分格式進行求解,接著研究了解的先驗估計式和穩(wěn)定性.最后,用一個數(shù)值算例說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：1840023