Gudermannian Function And Its Application For Solving Nonlinear Evolution Equations

With the process of deeply exploring on many complex phenomena in the nature, people gradually realize linearity and simpleness are not always the basic aspect of the nature; Additionally, the linear relation sometimes is incompetently appl ied to tackle the problems people encounter in their production activity. Meanwhile, nonlinear concepts and methods have already started to penetrate into many fields of technology. So the deeply and systematically studying on nonlinearity seems very indispensable. The important significance of research on nonlinear problems is not only embodied in the nature science aspects, but also in social sciences and in philosophy. However, when one wants to theoretically and systematically research the nonlinear pr oblems, solving nonlinear equations would be involved. The purpose of studying Gudermannian function in this thesis is just ready for benefitting to solve nonlinear equations.This thesis is mainly consisting of three parts below:In the first part, introducing the background of the nonlinear physics and the typical nonlinear phenomena and the nonlinear equation,in order to do groundwork for Gudermannian function and its application.The second part: Described Guderman and Gudermannian function and its characteristics including the formula, hyperbola, image and so on.The third part: On the basis of the thought of the least square method and using semi-analytical method, we not only constructed the Gudermannian function solution of nonlinear evolution equations but also verified its accuracy and feasibility.Based on the work, we find the analytical solution of nonlinear evolution equations which is calculated from Gudermannian function is good agreement with the accurate solution; this fully indicates the practical utility and validation of Gudermannian function in solving nonlinear equations. Additionally, although theoretically the solution of nonlinear equations expressed in the form of hyperbolic function always can be transformed into the formation of Gudermannian function, the kink solitary wave expressed in the form of Gudermannian function is more concise.