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, Dαa+ (a ⊂ R, 0 < α < 1) [equation presented] where {ak (x)} n-1k=0 are continuous real functions defined in [a, b] ⊂ R and [equation presented] 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 eαλx, which we will call a-exponential. 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. © 2007 Springer.