混合边界条件下广义二维多项时间分数阶扩散方程的解析解

Analytical solution of the generalized muti-term time-fractional diffusion equation in two-dimensions with mixed boundary condition

广义多项时间分数阶扩散方程已被用于描述一些重要的物理现象,目前,有关该类方程在高维情形下满足混合边界条件的研究仍较少.利用分离变量法考虑有界区域上广义二维多项时间分数阶扩散方程,方程中关于时间变量的分数阶导数采用Caputo分数阶导数的定义,其阶分别定义在[0,1],[1,2].而关于空间变量的偏导数则定义为传统的整数阶导数(二阶),得到了有界区域上广义二维多项时间分数阶扩散方程满足非齐次混合边界条件的解析解.亦可用于求解其他类型的满足不同边界条件的分数阶微分方程的解析解.

Generalized multi-term time-fractional diffusion equations have been used to describe important physical phenomena.However,studies on multi-term time-fractional diffusion equations with mixed boundary conditions in high dimensional conditions are still limited.In this paper,a method of separating variables was effectively implemented to solve a generalized multi-term time-fractional diffusion equation（GMTDE）in a finite domain.In this equation,the multi-term time-fractional derivatives were defined in the Caputo sense,whose orders belonged to the intervals[0,1],[1,2],respectively.The space partial derivatives were classical integer order derivatives whose order were 2.We discussed and derived the analytical solution of the GMTDE in two dimensions meeting nonhomogeneous mixed boundary conditions.The technique reported can be applied to other kinds of fractional differential equations with different boundary conditions.