The Generalized Solution for Transitional Shock Layer to Singularly Perturbed Nonlinear Hyperbolic Type Differential System with Two Parameters

DOI：10.16205/j.cnki.cama.2019.0020

 作者 单位 E-mail 韩祥临 湖州师范学院求真学院, 浙江 湖州 313000. xihan@zjhu.edu.cn 莫嘉琪 安徽师范大学数学与统计学院, 安徽 芜湖 241003. mojiagi@mail.ahnu.edu.cn

研究了一类两参数双曲型微分系统奇异摄动初始边值问题. 首先, 利用奇异摄动理论和方法, 注意到两个小参数, 构造了问题的外部解. 其次, 利用多重尺度变量和伸长变量, 分别得到了原问题解的过渡冲击层、边界层和初始层校正项. 最后, 得到了原问题解的渐近展开式, 并利用泛函分析不动点理论, 证明了渐近解的一致有效性. 由本方法求得的原问题的渐近解, 它还可以进行微分, 积分等解析运算, 从而能了解相应过渡冲击层解的更进一步的性态. 因此本方法具有良好的应用前景.

A class of nonlinear hyperbolic type differential system to singularly perturbed initial-boundary problem with two parameters is studied. Firstly, using singular perturbation theory and method, the outer solution for the problem is structured related two small parameters. Secondly, using the multi-scale and stretched variables, the transitional shock layer, boundary layer and initial layer corrective terms are obtained for the original problem respectively. Finally, the asymptotic expansion of solution for the original problem is given. And the uniform validity of its asymptotic solution is proved by using the theory of fixed point of functional analysis. Using this method to obtained asymptotic solution of original problem, it can also carry on analytical operation for the differential and integral and so on. It is known more behaviors for the transitional shock layer of solution. Thus this method possesses good applied foreground.