【摘要】：幾何不變流的研究來(lái)源于圖像處理和晶體增長(zhǎng)等方面,有著廣泛的應(yīng)用.本文運(yùn)用Lie對(duì)稱群方法系統(tǒng)地研究了兩個(gè)曲線流 中心仿射不變流和雙曲型仿射不變流的群不變解問(wèn)題.全文共有四章:第一章:介紹了研究背景及相關(guān)預(yù)備知識(shí),給出了本文的主要工作.第二章:研究了中心仿射不變流對(duì)應(yīng)的非線性偏微分方程的群不變解問(wèn)題.首先利用Lie群理論,給出了方程的對(duì)稱群.接著應(yīng)用Olsiannikov和Olver的思想方法,得到了一個(gè)最優(yōu)系統(tǒng)及其約化方程,最后討論了相應(yīng)的群不變解.第三章:我們研究了雙曲型仿射不變流對(duì)應(yīng)的非線性偏微分方程,計(jì)算了方程的李點(diǎn)對(duì)稱群,并構(gòu)造其一個(gè)最優(yōu)系統(tǒng),最后利用最優(yōu)系統(tǒng)對(duì)非線性偏微分方程進(jìn)行對(duì)稱約化,得到相應(yīng)的約化方程和一些群不變解.第四章:對(duì)全文進(jìn)行總結(jié).
[Abstract]:The research of geometric invariant flow comes from image processing and crystal growth and has been widely used. In this paper, we systematically study the problem of group invariant solutions for two central affine invariants and hyperbolic affine invariants by using lie symmetric group method. There are four chapters in this paper: chapter 1: introduce the research background and related preparatory knowledge, and give the main work of this paper. In chapter 2, the problem of group invariant solutions of nonlinear partial differential equations corresponding to central affine invariant flow is studied. Firstly, the symmetric group of the equation is given by using lie group theory. Then, by using Olsiannikov and Olver's method, an optimal system and its reduced equations are obtained. Finally, the corresponding group invariant solutions are discussed. In chapter 3, we study the nonlinear partial differential equation corresponding to hyperbolic affine invariant flow, calculate the lie point symmetric group of the equation, and construct an optimal system. Finally, we use the optimal system to reduce the nonlinear partial differential equation symmetrically. The corresponding reduction equation and some group invariant solutions are obtained. Chapter four: summarize the full text.
【學(xué)位授予單位】：寧波大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 ;Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation[J];Chinese Annals of Mathematics(Series B);2012年02期

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本文編號(hào)：2119466