近似对称约化与双曲函数方法的应用

【作者】 张睿；

【导师】 张顺利；

【作者基本信息】 西北大学 ， 应用数学， 2011， 硕士

【摘要】 随着当代科学技术的进步与发展,非线性科学逐步成为各个交叉学科与跨学科的关键,并且对于现代科技起到了决定与推进作用,为工业生产与技术革新提供了理论支持。许多物理学中与其他学科中的问题现在往往以非线性偏微分方程的形式来进行描述,因此,研究各学科内的非线性问题变得越来越重要,本文引入了近似约化方法与双曲函数方法对给定方程的近似解与精确解分别进行研究。第一章,首先介绍了近似对称方法的发展史,由此引出了关于解决扰动问题的扰动定理,结合扰动定理简单介绍了对带有弱扰动项进行对称约化的近似对称方法,对于一般情况下不含有小参数的扰动方程,介绍了同伦分析方法并结合此方法介绍了近似同伦对称方法。其次引入了双曲函数方法的历史背景以及这种方法对于研究一大类演化方程的精确解的意义。第二章,介绍了拓展双曲函数方法并且引入KP方程,引入了行波解的概念以及平衡方程的方法,从而对双曲函数方法进行进一步的延拓,运用延拓的双曲函数方法,进一步求得关于KP方程的多种精确解。第三章,介绍了K(n,m)方程并由此引出带有阻尼项的K(n,1)方程,应用线性同伦模型将阻尼K(n,1)方程重写并利用扰动定律将其化为方程组,然后分别应用对称方法与直接方法对所得到的方程组进行约化并求得其近似解,并对两种方法所得出的结果进行相互的推导。更多还原

【Abstract】 With progress and development of the modern science and technology, nonlinear science has gradually become a cross-discipline. Modern technology has played a pivotal role in providing a theoretical support to advance the technological innovation for industrial production. Physics and many other disciplines are now usually in the state of describing the issues in their field via the models of the nonlinear partial differential equations. Therefore, it becomes increasingly important to investigate the various nonlinear problems involved in these disciplines. We introduce the approximate reduction method and the hyperbolic function method to study approximate solutions and exact solutions of the given equations, respectively.The first chapter introduces the development of approximate symmetry method, which leads to the perturbed problems based on the perturbation theory. Combined with a brief introduction of the perturbation theory, weak perturbation symmetry reduction method is introduced. For the perturbed equations which do not contain a small parameter, we then introduce the approximate homotopy method. Moreover, we introduce the historical background of the hyperbolic function method.The second chapter describes the extended hyperbolic function method and the KP equations. We also introduce the concept of traveling wave solutions and the method of balancing the equation. Via the hyperbolic function method, we can further obtain the exact solutions of a variety of evolution equations.The third chapter introduces the K(n,l) equation which leads to the K(n,l) equation with damping. We rewrite the K(n,l) equation as a system of equations with damping via the linear homotopy model combined the law of perturbation theory. Then the symmetry and the direct method were applied to the transformed system to obtain its approximate solution. The correspondence of the results is also derived.更多还原