Toward a new understanding of cohomological method for fractional partial differential equations

One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation‎ ‎$ psi^{alpha} f=g $‎, ‎on‎ ‎$ mathbb{T}^n $‎. ‎We prove that if‎ ‎$ f $‎ ‎is not analytic‎, ‎then fractional integro-differential equation‎ ‎$ I_t^{1-alpha} D_x^{alpha}u(x,t)+i I_x^{1-alpha} D_t^{alpha}u(x,t)=f(t) $‎ ‎has no solution in‎ ‎$ C^1(B) $ with $0< alpha leq 1$‎. ‎‎W‎e ‎also‎ obtain ‎solutions ‎for‎ the space-time fractional heat ‎equations‎ on‎ ‎$ mathbb{S}^1 $‎ ‎and ‎$ mathbb{T}^n $‎. ‎At the end of this article‎, ‎there are examples of fractional partial differential equations and a fractional integral equation together with their solutions‎.