Analysis of solutions of some multi-term fractional Bessel equations

Abstract

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space $C^p$ is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation $$ \sum_{i=1}^{m}d_i\, x^{\alpha_i}D^{\alpha_i} u(x) + (x^\beta - \nu^2)u(x)=0, \ \ \alpha_i > 0, \beta > 0, $$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients $d_i$. Our theory leads to a threshold admissible value for $\nu^2$, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.