一类含时间分数阶导数的膜振动方程近似解析解的研究

Approximate analytical solution to a class of membrane vibration equations with time fractional derivatives

研究一类含时间分数阶导数的膜振动方程,该方程边界正弦摄动变化。先对边界自变量应用泰勒级数展开,引入多重尺度到原方程及边界,利用Riemann-Liouville分数阶导数定义和性质得到关于小参数的零阶近似解。应用微分不等式理论证明了解的一致有效性。利用图形分析出各参数对解的影响。

Here,a class of membrane vibration equation with time fractional derivative was studied.The equation boundary varied with sinusoidal perturbation.Taylor series was applied to expand the boundary independent variable,and then multi-scale were introduced to the original equation and boundary.By using the definition and properties of Riemann-Liouville fractional derivative,the approximate solution of the equation for the zero-order small parameter was obtained.Furthermore,the consistent effectiveness of the solution was proved by using the theory of differential inequalities.Finally,the influence of each parameter on the solution was analyzed using graphs.