Linear Differential Equations of Fractional Order

Abstract

This manuscript presents the basic general theory for sequential linear fractional differential equations, involving the well known Riemann-Liouville fractional operators, D a+ (a ∈ R, 0 < α ≤ 1) Lnα(y) = " D a+ + n−1 X k=0 ak(x)D a+ # (y) = y + n−1 X k=0 ak(x)y (kα = f(x). (1) where {ak(x)}n−1 k=0 are continuous real functions defined in [a, b] ⊂ R and D a+ = D a+ D a+ = D a+D a+ . (2) We also consider the case where f(x) is a continuous real function in (a, b] ⊂ R and f(a) = o(xα−1). We then introduce the Mittag-Leffler type function e α , which we will call αexponential. This function is the product of a Mittag-Leffler function and a power function. This function allows us to directly obtain the general solution to homogeneous and non-homogeneous linear fractional differential equations with constant coefficients. This method is a variation of the usual one for the ordinary case.