分数阶非线性薄膜方程的不变子空间和精确解

Invariant Subspace and Exact Solutions of Fractional Thin Film Equations

方程的精确解是方程的非线性现象以及它本身所蕴含的物理意义的一种具体表现,而不变子空间则可以理解为方程的稳定区域.在给出一类非线性薄膜方程所对应的几种不变子空间后,通过两边系数比较得到了其相应的有限维动力系统.再借助Mittag-Leffler函数的一些基本性质以及拉普拉斯变换等求解该系统,从而构造得到了分数阶非线性薄膜方程的一些经典类型的精确解,如指数型解、三角函数型解、多项式型解.

The exact solution of the equation is a concrete expression of the nonlinear phenomenon of the equation and its inherent physical meaning,and the invariant subspace can be understood as the stable range of the equation.After giving the invariant subspaces corresponding to nonlinear thin film equations,the corresponding finite dimensional dynamic systems are obtained by comparing the coefficients on both sides.Then,by means of some basic properties of Mittag-Leffler function and Laplace transform,some classical exact solutions of fractional nonlinear thin film equations are constructed,such as exponential solution,trigonometric solution and polynomial solution.