Numerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutions

‎In this paper‎, ‎we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem‎. ‎In ‎this ‎equation ‎the ‎time ‎derivative ‎is ‎the ‎Caputo ‎of ‎order ‎‎$‎‎gammain(0,1)$ ‎and ‎the ‎space ‎derivative ‎is ‎the ‎Riesz ‎of ‎order ‎‎$‎‎alphain(1,2]$.‎ ‎The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed‎ ‎schemes‎. W‎e prove that the new difference scheme is unconditionally stable‎. Also, ‎we find that the difference scheme is convergent with order $min{2-gamma,rgamma}$ in time for all $gammain (0,1)$ and $alpha in (1,2]$‎. ‎A test example is given to verify the efficiency and accuracy of the difference scheme‎.