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.

中文翻译：

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分数拟贝塞尔方程级数解的构造与分析

在本文中，我们介绍分数拟贝塞尔方程

$$\begin{aligned} \sum _{i=1}^{m}d_i x^{\xi _i}D^{\alpha _i} u(x) + (x^\beta - \nu ^2) u(x)=0 \end{对齐}$$

并在分数级数解类中构建它们的存在论。为了找到序列的参数，我们推导出特征方程，它与不匹配参数\(\xi _i\ne \alpha _i\)的项无关。我们的方法使我们能够获得包括准欧拉方程在内的广泛的分数微分方程的新结果。作为一个特定的例子，我们展示了我们的方法如何适用于常数系数方程。理论结果在计算上是合理的。