α-Analytic solutions of some linear fractional differential equations with variable coefficients

Abstract

This paper investigates the solutions, around an ordinary point x 0 ∈ [ a, b] for fractional linear differential equations of the form: [ L n α ( y ) ] ( x ) = g ( x , α ) , where [ L n α ( y ) ] ( x ) = y ( n α ) ( x ) + ∑ k = 0 n - 1 a k ( x ) y ( k α ) ( x ) with α ∈ (0, 1]. Here n ∈ N, the real functions g( x) and a k ( x) ( k = 0, 1, … , n − 1) are defined on the interval [ a, b], and y ( nα) ( x) represents sequential fractional derivatives of order kα of the function y( x). This study is an extension of the corresponding works by Al-Bassam.