Solution of Euler-Type Non-Homogeneous Differential Equations with Three Fractional Derivatives

Abstract

The linear non-homogeneous ordinary differential equations with three left-hand sided Lioville derivatives of fractional order are considered. Using the direct and inverse Mellin transforms and the residue theory, explicit solutions of such equations are established in terms of the generalized hypergeometric Wright functions, of the generalized hypergeometric functions and of the Euler psi function. The corresponding results are deduced for ordinary differential equations of Euler type. Examples are given.

Mathematics Subject Classification: 34A05, 26A33, 44A99, 33C20, 33C99.

Key Words and Phrases: linear differential equations with Liouville fractional derivatives, ordinary differential equations, explicit solutions, Mellin transforms, generalized Wright function, generalized hypergeometric function, Euler psi function.