Exact Solutions for a Class of Variable Coefficients Fractional Differential Equations Using Mellin Transform and the Invariant Subspace Method

In this paper, we propose a class of variable coefficients fractional ordinary differential equations (FODEs). Using Mellin transform (MT), we have transformed this class into a functional equation which can’t be solved in general. So, we have selected many special cases of this functional equation that can be solved exactly. After solving these special cases of the functional equation and using the inverse MT, we obtained some exact solutions for the proposed class. The obtained solutions are given in the form of \(H\) $H$ -function and the Wright function. The results, as special cases, contain some special forms given in the literature. Also, the invariant subspace method (ISM) is utilized for solving a class of nonlinear fractional diffusion equations with variable coefficients. The solutions of this class of nonlinear fractional diffusion equations depend upon the solutions of the proposed class of FODEs.