Linearization of Emden Differential Equation via the Generalized Sundman Transformations

The Emden differential equation is one of the most widely studied and challenging nonlinear dynamics equations in literature. It finds applications in various areas of study such as celestial mechanics, fluid mechanics, Steller structure, isothermal gas spheres, thermionic currents and so on. Because of the importance of the equation, the method of generalized Sundman transformation (GST) as proposed by Nakpim and Meleshko is used for linearizing the Emden differential equation. The Emden differential equation considered here is a modification of the equation given by Berkovic. The results obtained in this paper imply that the Emden equation cannot be linearized by a point transformation. The general solution of the modified Emden equation is also obtained.

基于特殊函数法的Rayleigh-Plesset方程理论解研究

Rayleigh-Plesset方程是气泡动力学理论中的基本方程,描述了无限不可压缩流体中,球形气泡的运动,求解Rayleigh-Plesset方程对于进一步深入研究气泡生长、发展和溃灭具有重要意义。该文主要结合Sundman幂变换和特殊函数理论分析了无黏蒸汽泡的Rayleigh-Plesset方程的理论解。研究表明:蒸汽泡动力学演化规律可由超几何函数形式和Weierstrass椭圆函数形式解析表达,当不考虑表面张力时,两种函数均可表示蒸汽泡的演化规律;当考虑表面张力时,蒸汽泡的动力学演化可由部分Weierstrass椭圆函数表达。该文给出了蒸汽泡在两种情况下的参数解析解具体形式,结果表明:常压载荷下的蒸汽泡只能发生一次溃灭,且蒸汽泡的溃灭时间随流体表面张力的增大而减小。