Akbar Dehghan Nezhad - Department of Mathematics, Iran University of Science and Technology, P.O.Box, ۱۶۸۴۶-۱۳۱۱۴, Narmak, Tehran, Iran.
Mina Moghaddam - Department of Mathematics, Iran University of Science and Technology, P.O.Box, ۱۶۸۴۶-۱۳۱۱۴, Narmak, Tehran, Iran.

One of the aims of this article is to investigate the solvability and unsolvability conditions for fractional cohomological equation ψ αf = g, on T n. We prove that if f is not analytic, then fractional integro-differential equation I ۱−α t Dα x u(x, t) + iI۱−α x Dα t u(x, t) = f(t) has no solution in C۱ (B) with ۰ < α ≤ ۱. We also obtain solutions for the space-time fractional heat equations on S ۱ and 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.

. Fractional calculus, Fractional cohomological equations, Space-time-fractional heat equation, Solvable and unsolvable fractional differential equations