Linear fractional differential equations with variable coefficients

This work is devoted to the study of solutions around an αα-singular point x0∈[a,b]x0∈[a,b] for linear fractional differential equations of the form [Lnα(y)](x)=g(x,α), where [Lnα(y)](x)=y(nα)(x)+∑k=0n−1ak(x)y(kα)(x) with α∈(0,1]α∈(0,1]. Here n∈Nn∈N, the real functions g(x)g(x) and ak(x)(k=0,1,…,n−1) are defined on the interval [a,b][a,b], and y(nα)(x)y(nα)(x) represents sequential fractional derivatives of order kαkα of the function y(x)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