發(fā)布時(shí)間：2018-05-01 12:10

  本文選題：Qiao方程 + Gardner-KP方程　； 參考：《昆明理工大學(xué)》2017年碩士論文

【摘要】：非線性演化方程在數(shù)學(xué)、物理學(xué)、化學(xué)、流體力學(xué)、振動(dòng)力學(xué)、天體力學(xué)、生物學(xué)、生態(tài)學(xué)和財(cái)政金融等自然科學(xué)和社會(huì)科學(xué)領(lǐng)域有著廣泛的應(yīng)用.一大批數(shù)學(xué)和其他領(lǐng)域科學(xué)工作者長(zhǎng)期以來(lái)致力于非線性演化方程的研究,在獲得了豐碩成果的基礎(chǔ)上不斷推進(jìn)非線性問(wèn)題的持續(xù)發(fā)展,為人類社會(huì)的生產(chǎn)、生活和科學(xué)研究提供了具有重要指導(dǎo)意義和應(yīng)用價(jià)值的結(jié)果.本文應(yīng)用經(jīng)典的李對(duì)稱分析方法,研究?jī)深惙蔷�性演化方程的解析解和守恒律問(wèn)題.李對(duì)稱分析方法是求解非線性偏微分方程和計(jì)算方程守恒律的重要理論之一,本文首先介紹了該方法的產(chǎn)生背景和基礎(chǔ)理論,給出了論文后續(xù)研究過(guò)程中用到的一系列關(guān)鍵定理和公式.本文第二部分重點(diǎn)研究Qiao方程,用李對(duì)稱分析求出了Qiao方程的無(wú)窮小生成子,選擇無(wú)窮小生成子的兩種線性組合對(duì)原方程進(jìn)行對(duì)稱約化,獲得了對(duì)應(yīng)的約化方程,并求出了群不變解.在群不變解的基礎(chǔ)上,通過(guò)采用一種新的構(gòu)造方式,獲得了一批原方程的非群不變解,并通過(guò)數(shù)值模擬畫(huà)圖對(duì)比了群不變解和非群不變解的區(qū)別.論文根據(jù)特殊的對(duì)稱,還求了方程的迭代解.最后對(duì)應(yīng)于每個(gè)生成子求出了Qiao方程的非局部守恒律公式.本文第三部分研究了Gardner-KP方程.基于李對(duì)稱分析,首先求出了方程的部分群不變解和迭代解.由于方程有三個(gè)獨(dú)立變量,一次對(duì)稱約化的方程仍然是偏微分形式,求解較為困難,為此對(duì)約化方程再次利用對(duì)稱分析將方程化為各種形式的常微分方程,并采用冪級(jí)數(shù)方法,求出常微分方程的冪級(jí)數(shù)解且證明了解的收斂性,從而獲得了一批原方程的冪級(jí)數(shù)解.對(duì)應(yīng)于第一次約化的每個(gè)生成子,本章還求出了相應(yīng)的非局部守恒律公式.論文最后對(duì)研究方法、過(guò)程和研究結(jié)果進(jìn)行了必要的總結(jié),并提出了下一步研究工作的目標(biāo)和方向.
[Abstract]:Nonlinear evolution equations are widely used in the fields of mathematics, physics, chemistry, fluid dynamics, vibration dynamics, astromechanics, biology, ecology, finance and finance. For a long time, a large number of scientists in mathematics and other fields have devoted themselves to the study of nonlinear evolution equations, and on the basis of obtaining fruitful results, they have continuously promoted the sustainable development of nonlinear problems for the production of human society. Life and scientific research provide important guiding significance and application value of the results. In this paper, the classical lie symmetry analysis method is used to study the analytical solutions and conservation laws of two kinds of nonlinear evolution equations. The lie symmetry analysis method is one of the important theories for solving nonlinear partial differential equations and the conservation law of computing equations. In this paper, the background and basic theory of the method are introduced. A series of key theorems and formulas are given. In the second part of this paper, the Qiao equation is studied, and the infinitesimal generator of Qiao equation is obtained by using the lie symmetry analysis. Two linear combinations of the infinitesimal generator are selected to reduce the original equation and the corresponding reductive equation is obtained. The group invariant solution is obtained. On the basis of group invariant solution, a group of nongroup invariant solutions of the original equation are obtained by adopting a new construction method, and the difference between group invariant solution and nongroup invariant solution is compared by numerical simulation drawing. According to the special symmetry, the iterative solution of the equation is also obtained. Finally, the nonlocal conservation law formula of Qiao equation is obtained for each generator. In the third part of this paper, we study the Gardner-KP equation. Based on lie symmetry analysis, the partial group invariant solution and iterative solution of the equation are first obtained. Because the equation has three independent variables, the equation of the first degree symmetry reduction is still a partial differential form, so it is difficult to solve the equation. Therefore, the reduced equation is transformed into ordinary differential equation of various forms by symmetric analysis again, and the power series method is adopted. The power series solutions of ordinary differential equations are obtained and the convergence of the solutions is proved, and the power series solutions of some original equations are obtained. For each generator of the first reduction, the corresponding nonlocal conservation law formula is also obtained in this chapter. Finally, the paper summarizes the research methods, processes and results, and puts forward the goal and direction of the next research.
【學(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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