Explicit Representations of Green's Function for Linear Fractional Differential Operator with Variable Coefficients

Abstract

We provide explicit representations of Green's functions for general linear fractional differential operators with {\it variable coefficients} and Riemann-Liouvilles derivatives. We assume that all their coefficients are continuous in $[0, \infty)$. Using the explicit representations for Green's function, we obtain explicit representations for solution of inhomogeneous fractional differential equation with variable coefficients of general type. Therefore the method of Green's function, which was developed in previous research for solution of fractional differential equation with constant coefficients, is extended to the case of fractional differential equations with {\it variable coefficients}.