【作者】 唐晓芬；

【导师】 蔡国梁；

【作者基本信息】 江苏大学 ， 应用数学， 2009， 硕士

【摘要】 在齐次平衡原则的思想下,充分利用F-展开法、双曲正切函数法和Riccati方程在非线性偏微分方程(NLPDES)求解中的优良特性,提出一种广义扩展的F-展开法和改进的双曲正切函数法。此方法在借助于计算机符号系统Mathematica下,操作方便,可以得到NLPDES的一系列精确解如周期波解、类孤子解、三角函数解、有理函数解、复数形式解,扭曲状解等。并利用广义扩展的F-展开法求解了非线性色散耗散mKdV方程,利用改进的双曲正切函数法求解了(3+1)维Burgers方程,得到了它们类型丰富的精确解,其中部分是新解。文中对部分解进行了数值模拟以便直观分析。首先,在齐次平衡思想的基础上利用改进的辅助方程方法求解Klein-Gordon方程,得到了更多丰富类型的Klein-Gordon方程的行波解如类孤子解,三角函数周期解,有理数解,指数解。其次,利用广义扩展的F-展开法研究了非线性色散耗散mKdV方程,得到了非线性色散耗散mKdV方程的周期波解、类孤子解、三角函数解、有理函数解、复数形式解等。这些解对于解释一些物理现象具有一定的意义。最后,利用改进的双曲正切函数法研究了(3+1)维Burgers方程,得到了他们类型丰富的精确解:光滑的钟形孤立波解,kink解,类孤子解,复数形式解,有理数解等,并得到了部分新解。这对于对非线性偏微分方程的进一步研究具有积极的意义。更多还原

【Abstract】 Under the Homogeneous balance idea, the generalized modified F-expansion method and the extended modified tanh-function method are proposed by taking full advantages of F-expansion method, tanh-function method and Riccati equation in seeking exact solutions of NLPDEs. The method can be conveniently operated with the aid of computer symbolic systems Mathematica, and rich families of exact solutions of NLPDEs have been obtained, including periodic wave solutions, solitary wave solutions, triangle function solutions, rational function solutions, plural number formal solutions, bell-shaped solitary solutions, kink-shape solutions and so on. By using the generalized modified F-expansion method and the extended modified tanh-function method, we have solved the nonlinear dispersive dissipative mKdV equation and the (3+1)-dimensional Burgers equations respectively. Massive exact solutions of them have been obtained, and some of the solutions are new. We also provided some figures of partial solutions for direct-viewing analysis.Firstly, we researched the Klein-Gordon equation by using the modified auxiliary method under Homogeneous balance idea. As a result, many new and more general exact traveling wave solutions are obtained, such as soliton-like solutions, trigonometric function solutions, exponential solutions and rational solutions, etc.Next, we researched the exact solutions of the nonlinear dispersive dissipative mKdV equation by using a generalized modified F-expansion method. Rich families of exact solutions of them have been obtained, including periodic wave solutions, soliton-like solutions, triangle function solutions, rational function solutions, plural number formal solutions and so on. And some of them are new. We consider these solutions will make sense for explaining some physical phenomenon.Finally, we used the extended modified tanh-function method to solve the (3+1)-dimensional Burgers equations and obtained more rich exact solutions of them, like bell-shaped solitary solutions, kink solutions, soliton-like solutions, plural number formal solutions, rational solutions and so on. This work will be usefull for further research to the equation.更多还原