一类含分数阶阻尼的三维波导中的传播问题  

Propagation Problem about a Three-dimensional Wave with Fractional Damping

作　　者：葛志新[1] 李春源 陈咸奖 GE Zhi-xin;LI Chun-yuan;CHEN Xian-jiang(School of Mathematics&Physics,Anhui University of Technology,Ma'anshan 243002;School of Economics,Anhui University of Technology,Ma'anshan 243002)

机构地区：[1]安徽工业大学数理学院,马鞍山243002 [2]安徽工业大学商学院,马鞍山243002

出　　处：《工程数学学报》2021年第3期389-398,共10页Chinese Journal of Engineering Mathematics

基　　金：安徽省高校自然科学研究重点项目(KJ2016A084,KJ2019A0062);2018年安徽工业大学大学生创新创业训练计划项目推荐项目(省级)(201810360367)。

摘　　要：本文研究一类三维波动方程,该方程含有分数阶小阻尼,边界含有小参数,并做正弦波动.我们利用多重尺度方法和Riemann-Liouville分数阶导数的定义及性质,对原边值问题应用泰勒公式,得到关于小参数的零阶和一阶方程边值问题.利用分离变量法,引入解谐参数,通过分析边值问题的可解性条件得到零阶近似解的振幅和相位的变化规律.然后,用微分不等式证明了解的一致有效性.最后分析了该问题二维波与三维波解的区别,并通过图形展示了三维波振幅关于相关参数的变化规律.这个三维波动边值问题说明,当边界发生正弦型小波动,垂直这个边界上的外力有规律地变化,则该波有一个近似解,该解的振幅的模和相位的瞬时变化率由边界取值、最初选择的模态值和分数阶导数的取值确定.可以发现没有阻尼二维波与三维波的解的振幅有很大差异.二维波仅仅是振幅相位在周期变化,振幅模却恒定,近似解是周期解.该三维波是振幅模和相位两者都在变化,但小参数对波动影响不大.In this paper,a class of three-dimensional wave equations with small fractional damping and sine-waving on the boundary is investigated.The boundary of the problem contains small parameters.Using the multi-scale method and the definition and properties of the Riemann-Liouville fractional derivative,the Taylor formula is applied to the original boundary value problem.The zero-order and first-order boundary value problems with respect to small parameters are obtained.Using the method of separating variables,introducing the detuning parameters,and analyzing the solvability conditions of the boundary value problem,the amplitude and phase of the zero-order approximate solution are obtained.Then,the uniformly valid behavior of the solution is illustrated by the theory of differential inequalities.Finally,the difference between the two-dimensional wave solution and the three-dimensional wave solution is analyzed.The changes in the amplitude of the three-dimensional wave with respect to relevant parameters are shown.The three-dimensional fluctuation boundary value problem shows that,when the boundary has small sinusoidal fluctuations and the external force perpendicular to the boundary changes regularly,the wave has an approximate solution.The instantaneous rate of change of the amplitude mode and phase of the solution is determined by the boundary value,the initially mode value and the value of the fractional derivative.It can be found that the amplitudes of the solutions of undamped two-dimensional waves and those of three-dimensional waves are obviously different.The two-dimensional wave only changes the amplitude and phase periodically,but the amplitude mode is constant,and the approximate solution is periodic.The three-dimensional wave changes in both amplitude mode and phase,but small parameters have little effect on its fluctuation.

关 键 词：多重尺度 分数阶导数 三维波动方程 可解性条件 

分 类 号：O175.14[理学—数学]