本文選題：古德曼函數(shù)　切入點(diǎn)：最小二乘法　出處：《西北師范大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：隨著對自然界中各種復(fù)雜現(xiàn)象研究的深入,人們認(rèn)識到自然界不都是線性與簡單的統(tǒng)一,線性關(guān)系已經(jīng)不能完全處理人類生產(chǎn)活動(dòng)中遇到的問題。如此同時(shí),非線性的概念和方法已經(jīng)開始向科學(xué)技術(shù)的各個(gè)領(lǐng)域滲透;對非線性問題進(jìn)行深入研究顯得非常必要。非線性問題的研究不僅對自然科學(xué)意義重大,而且對社會(huì)科學(xué)、哲學(xué)也具有重要的意義。要對非線性問題理論化、系統(tǒng)化研究將涉及到非線性方程的求解,本文研究古德曼函數(shù)的目的就是為了能更好地求解非線性方程做準(zhǔn)備。本文用三部分介紹古德曼函數(shù)及其應(yīng)用:第一部分:敘述了非線性物理的背景和典型的非線性現(xiàn)象、方程,為研究古德曼函數(shù)及其應(yīng)用做鋪墊。第二部分:介紹古德曼以及古德曼函數(shù)的公式、恒等式,反函數(shù),余函數(shù)以及它們的圖像、特點(diǎn)、性質(zhì)等。第三部分:在最小二乘法的思想基礎(chǔ)上,用半解析方法,構(gòu)造非線性演化方程的古德曼函數(shù)解;并驗(yàn)證它的準(zhǔn)確性和可行性。通過以上的研究工作,我們發(fā)現(xiàn),利用古德曼函數(shù)給出的非線性演化方程的解析解與準(zhǔn)確解吻合的很好,充分顯示了古德曼函數(shù)對求解非線性方程的實(shí)用性和有效性,達(dá)到了預(yù)期效果。另外,理論上,能夠用雙曲函數(shù)表示的非線性方程的解都可以轉(zhuǎn)換為古德曼函數(shù)表示,并且從文中列出幾例用古德曼函數(shù)表示的非線性方程的解來看,用古德曼函數(shù)表示扭結(jié)孤立子波在形式上更簡潔。
[Abstract]:With the deepening of the study of various complex phenomena in nature, people realize that the nature is not always linear and simple, linear relations can no longer fully deal with the problems encountered in human production activities. At the same time, The concept and method of nonlinearity have begun to infiltrate into various fields of science and technology. It is very necessary to study the nonlinear problem. The study of nonlinear problem is of great significance not only to natural science, but also to social science. Philosophy is also of great significance. To theorize nonlinear problems, systematic research will involve solving nonlinear equations. In this paper, the purpose of studying the Goodman function is to prepare for the better solution of the nonlinear equation. This paper introduces the Goodman function and its applications in three parts: the first part: the background of nonlinear physics and the typical nonlinear phenomena. The second part introduces the formulas, identities, inverse functions, residual functions and their images of Goodman and its functions. The third part: on the basis of the least square method, we construct the solution of the Goodman function of the nonlinear evolution equation by using the semi-analytic method, and verify its accuracy and feasibility. The analytical solution of the nonlinear evolution equation given by the Goodman function is in good agreement with the exact solution, which fully shows the practicability and validity of the Goodman function for solving the nonlinear equation, and achieves the expected effect. The solutions of the nonlinear equations which can be expressed by hyperbolic functions can be transformed into the expression of the Goodman functions, and the solutions of the nonlinear equations expressed by the Goodman functions are listed in this paper. The expression of kink solitary wavelet by Goodman function is more concise in form.
【學(xué)位授予單位】：西北師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O175.29

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