求分数阶偏微分方程精确解的两类方法

发布时间：2018-01-18 03:31

本文关键词：求分数阶偏微分方程精确解的两类方法　出处：《江苏大学》2017年硕士论文　论文类型：学位论文

【摘要】：近年来,分数阶微分方程的应用范围已经涉及到了生物工程、高能物理、系统控制、反常扩散等诸多领域。然而关于分数阶微分方程的求解,目前并没有统一的方法。因此,对分数阶微分方程进行求解成为了一个热门的研究领域。本文介绍了分数阶微积分的相关概念,并依据修改的Riemann-Liouville导数定义将分数阶微积分的理论应用于分数阶微分方程的求解。文中对扩展的指数函数展开法进行了改进,并运用该方法和首次积分法分别求解了分数阶Sharma-Tasso-Olever(STO)方程,分数阶Cahn-Allen(CA)方程和分数阶Whitham-Broer-Kaup(WBK)方程组,得到了三角函数、双曲函数、有理函数、指数函数等各种类型的精确解。这些实例说明这两种求解分数阶微分方程的方法具有很好的有效性和简易性。
[Abstract]:In recent years, the application of fractional differential equations has been involved in many fields, such as bioengineering, high energy physics, system control, anomalous diffusion and so on. At present, there is no uniform method. Therefore, solving fractional differential equations has become a hot research field. In this paper, the concept of fractional calculus is introduced. The theory of fractional calculus is applied to the solution of fractional differential equations according to the modified Riemann-Liouville derivative definition. The extended exponential function expansion method is improved in this paper. The fractional Sharma-Tasso-OleverSTO equation is solved by using this method and the first integral method, respectively. The trigonometric function, hyperbolic function and rational function are obtained for the fractional Cahn-AllenCA equation and the fractional Whitham-Broer-Kaupn WBK equations. These examples show that the two methods for solving fractional differential equations are effective and simple.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2

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