本文選題：對稱 + 微分特征列集算法　； 參考：《內(nèi)蒙古工業(yè)大學(xué)》2017年碩士論文

【摘要】：自然科學(xué)和工程技術(shù)中的很多問題本質(zhì)上就是微分方程,而偏微分方程(組)(簡稱為PDEs)是微分方程研究的主體,特別是非線性PDEs(簡稱為NLPDEs),所以求解NLPDEs的研究具有重要的意義.由于非線性方程本身的比較復(fù)雜,所以求解具有一定的難度.為了求解PDEs人們提出了眾多求解方法,但還沒有統(tǒng)一而系統(tǒng)的方法包攬各種解的求解,并且這些方法具有各自的適用范圍.從而研究求解方法仍是數(shù)學(xué)、物理、力學(xué)學(xué)科中的基礎(chǔ)性問題,特別是現(xiàn)有方法的改進(jìn)、總結(jié)歸納、加深認(rèn)識、接納優(yōu)點(diǎn)、摒棄缺陷,尤為必要,是發(fā)現(xiàn)新方法的前提.眾多方法中Lie對稱是通用性最好的方法,它以眾多傳統(tǒng)方法為其特例.目前PDEs對稱理論和方法在數(shù)學(xué)、物理和力學(xué)等學(xué)科中得到了廣泛的應(yīng)用.本文將基于微分特征列集算法,對Lie對稱方法和對稱分類在NLPDEs邊值問題中的應(yīng)用進(jìn)行研究.具體研究內(nèi)容有:第一章,著重綜述了對稱方法的發(fā)展現(xiàn)狀和在PDEs的研究中的重要性,并介紹了微分特征列集算法、龍格-庫塔法和同倫攝動法.第二章,通過有效結(jié)合對稱方法和數(shù)值計(jì)算方法(即龍格-庫塔法),求解了一個流體力學(xué)中的NLPDEs邊值問題的數(shù)值解.第三章,研究對稱分類在NLPDEs邊值問題中的應(yīng)用,具體計(jì)算了2個流體力學(xué)中的NLPDEs邊值問題的對稱分類,并對其進(jìn)行了求解.步驟如下:(1)基于微分特征列集算法,分析確定了含參數(shù)的NLPDEs邊值問題的對稱分類,并根據(jù)方程參數(shù)的不同取值,分類確定方程的主對稱和擴(kuò)充對稱.(2)利用確定的擴(kuò)充對稱將所研究的NLPDEs邊值問題約化為ODEs初值問題.(3)借助Mathmatica符號系統(tǒng),求解了ODEs初值問題的數(shù)值解.第四章,通過將對稱方法和近似解析解方法(即同倫攝動法)有效的結(jié)合,求解了2個NLPDEs邊值問題.先利用對稱方法把NLPDEs邊值問題約化為ODEs初值問題,再利用同倫攝動法對其進(jìn)行求解,得到了近似解.最后利用數(shù)值方法得到了數(shù)值解,并與近似解進(jìn)行比較,驗(yàn)證了近似解收斂于數(shù)值解.最后總結(jié)文章所研究的內(nèi)容,并對下一步的相關(guān)研究進(jìn)行了展望.
[Abstract]:Many problems in natural science and engineering technology are essentially differential equations, and partial differential equations (PDEs) (referred to as PDEs) are the main research subjects of differential equations, especially nonlinear PDEs (NLPDEs), so the study of solving NLPDEs is of great significance. Because of the complexity of nonlinear equations, it is difficult to solve them. In order to solve PDEs, many methods have been proposed, but there are no unified and systematic methods to solve all kinds of solutions, and these methods have their own scope of application. Therefore, it is necessary to study the solution method in mathematics, physics and mechanics, especially the improvement of the existing methods, to sum up, deepen the understanding, accept the advantages and abandon the defects, which is the premise of finding the new method. Lie symmetry is the most versatile method among many methods, and it takes many traditional methods as its special case. At present, PDEs symmetry theory and method have been widely used in mathematics, physics and mechanics. In this paper, the application of lie symmetry method and symmetric classification to NLPDEs boundary value problem is studied based on differential characteristic sequence set algorithm. The main contents are as follows: in the first chapter, the development of symmetric methods and their importance in the study of PDEs are reviewed, and the differential characteristic set algorithm, Runge-Kutta method and homotopy perturbation method are introduced. In chapter 2, the numerical solution of a NLPDEs boundary value problem in hydrodynamics is solved by combining the symmetric method with the numerical method (Runge-Kutta method). In chapter 3, the application of symmetric classification in NLPDEs boundary value problem is studied. The symmetric classification of two NLPDEs boundary value problems in hydrodynamics is calculated and solved. The steps are as follows: (1) based on the differential characteristic sequence set algorithm, the symmetric classification of NLPDEs boundary value problems with parameters is analyzed and determined according to the different values of the equation parameters. Classification determines the principal symmetry and extended symmetry of the equation. (2) the NLPDEs boundary value problem is reduced to the ODEs initial value problem by using the deterministic extended symmetry. (3) the numerical solution of the ODEs initial value problem is solved by means of the Mathmatica symbolic system. In chapter 4, two NLPDEs boundary value problems are solved by combining the symmetric method with the approximate analytical solution method (that is, the homotopy perturbation method). The NLPDEs boundary value problem is reduced to an ODEs initial value problem by using the symmetric method, and the approximate solution is obtained by using the homotopy perturbation method. Finally, the numerical solution is obtained by numerical method, and compared with the approximate solution, it is verified that the approximate solution converges to the numerical solution. Finally, the paper summarizes the content of the study, and prospects for the next related research.
【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

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