Construction and analysis of series solutions for fractional quasi-Bessel equations

Abstract

In this paper we introduce fractional quasi-Bessel equations $$\begin{aligned} \sum _{i=1}^{m}d_i x^{\xi _i}D^{\alpha _i} u(x) + (x^\beta - \nu ^2)u(x)=0 \end{aligned}$$and construct their existence theory in the class of fractional series solutions. In order to find the parameters of the series, we derive the characteristic equation, which is surprisingly independent of the terms with non-matching parameters \(\xi _i\ne \alpha _i\). Our methodology allows us to obtain new results for a broad class of fractional differential equations including quasi-Euler equations. As a particular example, we demonstrate how our approach works for the constant-coefficient equations. The theoretical results are justified computationally.