Fractional differential equations solved by using Mellin transform

•We find the solution of fractional different equations by using Mellin transform.•The shift in the argument of the transformed fractional operators is overcame.•Solution is found by solving an algebraic set of linear equations.•Self-similarity of the inverse Mellin transform and related Hurst exponent is found.

In this paper, the solution of the multi-order differential equations, by using Mellin transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.