Linear fractional differential equations with variable coefficients

Abstract

This work is devoted to the study of solutions around an α -singular point x 0 ∈ [ a , b ] for linear fractional 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 ( n α ) ( x ) represents sequential fractional derivatives of order k α of the function y ( x ) . This study is, in some sense, a generalization of the classical Frobenius method and it has applications, for example, in obtaining generalized special functions. These new special functions permit us to obtain the explicit solution of some fractional modeling of the dynamics of many anomalous phenomena, which until now could only be solved by the application of numerical methods. 1 1 This research was funded, in part, by MEC (MTM2004-00327 and MTM2007-60246) and by ULL.