古德曼函数及其在解非线性方程中的应用

发布时间：2018-03-06 17:01

本文选题：古德曼函数　切入点：最小二乘法　出处：《西北师范大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着对自然界中各种复杂现象研究的深入,人们认识到自然界不都是线性与简单的统一,线性关系已经不能完全处理人类生产活动中遇到的问题。如此同时,非线性的概念和方法已经开始向科学技术的各个领域渗透;对非线性问题进行深入研究显得非常必要。非线性问题的研究不仅对自然科学意义重大,而且对社会科学、哲学也具有重要的意义。要对非线性问题理论化、系统化研究将涉及到非线性方程的求解,本文研究古德曼函数的目的就是为了能更好地求解非线性方程做准备。本文用三部分介绍古德曼函数及其应用:第一部分:叙述了非线性物理的背景和典型的非线性现象、方程,为研究古德曼函数及其应用做铺垫。第二部分:介绍古德曼以及古德曼函数的公式、恒等式,反函数,余函数以及它们的图像、特点、性质等。第三部分:在最小二乘法的思想基础上,用半解析方法,构造非线性演化方程的古德曼函数解;并验证它的准确性和可行性。通过以上的研究工作,我们发现,利用古德曼函数给出的非线性演化方程的解析解与准确解吻合的很好,充分显示了古德曼函数对求解非线性方程的实用性和有效性,达到了预期效果。另外,理论上,能够用双曲函数表示的非线性方程的解都可以转换为古德曼函数表示,并且从文中列出几例用古德曼函数表示的非线性方程的解来看,用古德曼函数表示扭结孤立子波在形式上更简洁。
[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.
【学位授予单位】：西北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

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