Euler type differential equations of fractional order

Using the direct and inverse Mellin transforms, we find a solution to the nonhomogeneous Euler-type differential equation with Riemann-Liouville fractional derivatives on the half-axis in the class of functions represented by the fractional integral in terms of the fractional analogue of the Green's function. Fractional analogues of the Green's function are constructed in the case when all roots of the characteristic polynomial are different and in the case when there are multiple roots of the characteristic polynomial. Theorems of solvability of the nonhomogeneous fractional differential equations of Euler type on the half-axis are stated and proved. Special cases and examples are considered.