Abstract This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence of the proposed method are investigated then the convergence rate O ( h k + 1 + △ t 2 − α ( t n ) ) in the case of sub-diffusion and O ( h k + 1 + △ t ) in the case of super-diffusion are proven for the presented scheme. Finally, provided numerical examples illustrate efficiency of the method and accuracy of the theory.

中文翻译：

---

具有亚扩散和超扩散的时变阶分数阶微分方程的局部不连续Galerkin方法

摘要 本文提出了具有亚扩散和超扩散的时变阶分数阶微分方程的近似解。论文的目的是通过一个完全离散的局部不连续伽辽金方案来解决和分析这个问题。该方法基于空间上的局部不连续伽辽金方法和时间上的有限差分技术。研究了所提出方法的数值稳定性和收敛性，然后研究了子扩散情况下的收敛速度O ( hk + 1 + △ t 2 − α ( tn ) ) 和情况下的O ( hk + 1 + △ t )对于所提出的方案，证明了超扩散。最后，通过数值算例说明了该方法的有效性和理论的准确性。