带周期边界的时间分数阶扩散方程的差分格式

Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions

鉴于分数阶方程的解析解实难求得,本文主要研究了带周期边界的时间分数阶扩散方程的有限差分方法,时间方向采用L2-1σ离散公式,空间方向采用二阶差分格式离散,数值格式整体可达到二阶精度.随后利用Fourier方法证明了有限差分格式的唯一可解性、稳定性和收敛性.最后用MATLAB语言对具体的模型进行了数值求解,数值实验能很好地印证理论结果.

Generally,the analytic solution for fractional differential equations is hard to obtain.The difference scheme of time fractional diffusion equations with periodic boundaries is mainly studied,adopting the L2-1σ formula in temporal direction,and the second order difference approximation in spatial direction.The proposed scheme can obtain second order accuracy globally.The unique solvability,stability and convergence of the proposed scheme are proved by the Fourier method.Finally,the specific fractional models are solved numerically by MATLAB language.Numerical experiments are carried out to support the theoretical results.