Incommensurate multi-term fractional differential equations with variable coefficients with respect to functions

Abstract

Linear fractional differential equations with variable coefficients and various types of fractional derivatives can be solved explicitly, using either fixed-point theorems and successive approximations or Green's functions. In this paper, we consider a multi-term fractional differential equation with continuous variable coefficients and incommensurate fractional orders, and we construct an explicit solution by a direct method of successive approximations, which is made rigorous by checking the solution function explicitly via substitution. The results are obtained for the most general initial value problem of an inhomogeneous equation with inhomogeneous initial conditions, the first time that an explicit analytical solution has been found for this general problem. Special cases are provided to illustrate the main results, and connections are discussed with previous results in the literature.