Linear fractional differential equations with variable coefficients. (English) Zbl 1152.34305

Summary: This work is devoted to the study of solutions around an \(\alpha \)-singular point \(x_{0}\in [a,b]\) for linear fractional differential equations of the form \([\mathbf L_{n\alpha}(y)](x)=g(x,\alpha)\), where \[ [\mathbf L_{n\alpha}(y)] (x)= y^{(n\alpha)} (x) + \sum^{n-1}_{k=0} a_k (x) y^{(k\alpha)} (x) \] with \(\alpha \in (0,1]\). Here \(n \in N\), the real functions \(g(x)\) and \(a_k(x) (k=0,1,\dots,n-1)\) are defined on the interval \([a,b]\), and \(y(n\alpha)(x)\) represents sequential fractional derivatives of order \(k\alpha \) 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.


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
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