本文選題：時滯微分方程　切入點：數(shù)值解　出處：《哈爾濱工業(yè)大學》2015年碩士論文　論文類型：學位論文

【摘要】：本文對兩類時滯偏微分方程的差分方法展開研究,并且進行了理論分析。由于帶時滯量的微分方程的復雜性,大多數(shù)時滯微分方程都不能求得顯式的解析表達式。因此,此類方程的數(shù)值解法在理論和實際應用中都有著重要意義。近幾十年來,對時滯微分方程的數(shù)值處理的研究在國際上掀起了高潮,許多求解時滯微分方程的數(shù)值方法也陸續(xù)被提了出來,如Runge-Kutta法、?法、線性多步法等。對時滯微分方程的數(shù)值處理在自動控制、土木工程及環(huán)境科學等諸多領域中扮演著越來越重要的角色。本文首先對差分法的基礎知識與應用進行簡單的介紹。其次,針對一類含有小延遲量的偏微分方程初邊值問題,通過Taylor展開式的思想將其轉化為不帶延遲量的偏微分方程,然后再構造差分格式,并用已經(jīng)非常成形的沒有延遲項的偏微分方程的知識對格式進行理論分析。數(shù)值算例中通過與其他方法數(shù)值解的比較,得到本文中所介紹的數(shù)值方法具有較好的適用性,有更高的數(shù)值精度。最后,對另一種類型的時滯拋物型方程構造了Crank-Nicolson格式,經(jīng)理論分析發(fā)現(xiàn)它是一種無條件穩(wěn)定的差分格式。同樣的,通過數(shù)值算例對穩(wěn)定性進行了驗證。
[Abstract]:In this paper, the difference method for two kinds of partial differential equations with delay is studied, and the theoretical analysis is made. Because of the complexity of differential equations with delay, most delay differential equations can not obtain explicit analytical expressions. The numerical solution of this kind of equations is of great significance both in theory and in practice. In recent decades, the research on numerical treatment of delay differential equations has aroused a high tide in the world. Many numerical methods for solving delay differential equations have been proposed one after another, such as Runge-Kutta method? Method, linear multistep method, etc. The numerical processing of delay differential equation is controlled automatically, Civil engineering and environmental science play a more and more important role in many fields. In this paper, the basic knowledge and application of difference method are introduced briefly. Secondly, for a class of initial boundary value problems of partial differential equations with small delay, By using the idea of Taylor expansion, it is transformed into partial differential equation with no delay, and then the difference scheme is constructed. The scheme is theoretically analyzed by using the knowledge of partial differential equations with no delay term which has been formed very well. By comparing the numerical solutions with other numerical solutions, the numerical method presented in this paper has good applicability. Finally, Crank-Nicolson scheme is constructed for another type of delay parabolic equation, which is found to be an unconditionally stable difference scheme by theoretical analysis. Similarly, the stability is verified by numerical examples.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O241.82

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本文編號：1627503