本文選題：Lie對稱 + 最優(yōu)系統(tǒng)��； 參考：《內(nèi)蒙古工業(yè)大學(xué)》2017年碩士論文

【摘要】：隨著近代物理和數(shù)學(xué)的發(fā)展,物理學(xué)中的非線性現(xiàn)象、問題受到越來越多人的關(guān)注.許多非線性問題的研究可以被歸結(jié)為對非線性偏微分方程(以下簡稱PDE)的研究.求解非線性PDE是十分必要的,方法也有很多.Lie對稱方法是一個較普適性而行之有效的方法,是研究非線性PDE不變解的基礎(chǔ).本文基于符號計(jì)算系統(tǒng)MATHEMATICA,研究了一類非線性偏微分方程組(PDEs)和兩類非線性PDE的經(jīng)典Lie對稱、條件對稱、近似對稱、對稱分類、一維最優(yōu)系統(tǒng)、相似約化及不變解的構(gòu)造.第一章簡述了本文的研究背景及意義,并簡單介紹了經(jīng)典Lie對稱、條件對稱、近似Lie對稱的方法.第二章利用經(jīng)典Lie對稱的方法研究了一類含兩個任意函數(shù)的非線PDEs,獲得對稱分類.對其中的兩種情況做進(jìn)一步分析,構(gòu)造一維最優(yōu)系統(tǒng),并利用最優(yōu)系統(tǒng)中的元素對PDEs相似約化,求不變解.另外,利用條件對稱的方法研究了PDEs的一種特殊情況,并利用條件對稱對該方程組進(jìn)行相似約化、求不變解.第三章研究了一類非線性滲流方程vt=k(v_x)v_(xx).當(dāng)k(v_x)=e~x和k(v_x)=(v_x)n時,分別對這兩種情況的PDE進(jìn)行研究.構(gòu)造一維最優(yōu)系統(tǒng),對PDE進(jìn)行相似約化,進(jìn)而求不變解.此外,還分別利用條件對稱研究了這兩種情況的PDE,并利用條件對稱對方程進(jìn)行相似約化,進(jìn)而求不變解.第四章利用Baikov,Gazizov和Ibragimov提出的近似對稱方法,研究了擾動Boussinesq方程.構(gòu)造了一維最優(yōu)系統(tǒng),分析方程的近似不變解,并給出了一些近似不變量.第五章對本文的研究內(nèi)容進(jìn)行總結(jié),展望需要進(jìn)一步研究的內(nèi)容.
[Abstract]:With the development of modern physics and mathematics, more and more people pay attention to nonlinear phenomena in physics. Many studies on nonlinear problems can be attributed to the study of nonlinear partial differential equations (PDE). It is very necessary to solve nonlinear PDE, and there are many methods. Lie symmetry method is a more universal and effective method, which is the basis of studying nonlinear PDE invariant solution. In this paper, the classical lie symmetry, conditional symmetry, approximate symmetry, symmetric classification, one-dimensional optimal system, similarity reduction and invariant solution of a class of nonlinear partial differential equations (PDEs) and two classes of nonlinear PDE are studied based on the symbolic computing system MATHEMATICA. In the first chapter, the research background and significance of this paper are briefly introduced, and the classical lie symmetry, conditional symmetry and approximate lie symmetry are briefly introduced. In chapter 2, we use the classical lie symmetry method to study a class of nonlinear PDEs with two arbitrary functions, and obtain symmetric classification. The two cases are further analyzed to construct the one-dimensional optimal system, and the elements in the optimal system are used to reduce the PDEs similarity to obtain the invariant solution. In addition, a special case of PDEs is studied by using the method of conditional symmetry, and the equations are reduced similarly by using conditional symmetry, and the invariant solution is obtained. In chapter 3, we study a class of nonlinear seepage equation vt=k (VX) v _ (xx). When k (VX) and k (VX) = (vSX) n, the PDE of these two cases are studied respectively. The one-dimensional optimal system is constructed, and the PDE is reduced by similarity, and the invariant solution is obtained. In addition, the PDE of these two cases is studied by using conditional symmetry, and the equation is reduced similarly by using conditional symmetry, and the invariant solution is obtained. In chapter 4, the perturbation Boussinesq equation is studied by using the approximate symmetry method proposed by Baikovan Gazizov and Ibragimov. The one-dimensional optimal system is constructed, the approximate invariant solution of the equation is analyzed, and some approximate invariants are given. The fifth chapter summarizes the research content of this paper, and looks forward to further research content.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.29

【參考文獻(xiàn)】

相關(guān)期刊論文 前5條

1 蘇道畢力格;王曉民;烏云莫日根;;對稱分類在非線性偏微分方程組邊值問題中的應(yīng)用[J];物理學(xué)報(bào);2014年04期

2 鄭麗霞;郭華;白銀;;Boiti-Leon-Pempinelli方程組的相似約化及精確解[J];內(nèi)蒙古大學(xué)學(xué)報(bào)(自然科學(xué)版);2009年05期

3 張全舉,馮芙葉;廣義Burgers方程的非經(jīng)典相似約化[J];工程數(shù)學(xué)學(xué)報(bào);2003年06期

4 劉若辰,何文麗,張順利,屈長征;非線性發(fā)展方程的一維最優(yōu)系統(tǒng)(英文)[J];西北大學(xué)學(xué)報(bào)(自然科學(xué)版);2003年04期

5 朝魯;微分方程(組)對稱向量的吳-微分特征列算法及其應(yīng)用[J];數(shù)學(xué)物理學(xué)報(bào);1999年03期

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