Some estimates for Mittag-Leffler function in quantum calculus and applications

TL;DR

This paper establishes bounds for the $q$-Mittag-Leffler function and applies these estimates to analyze the solvability of time-fractional pseudo-parabolic equations within quantum calculus, expanding understanding of these special functions and their applications.

Contribution

The paper provides new bounds for the $q$-Mittag-Leffler function and uses them to study the solvability of fractional pseudo-parabolic equations in quantum calculus.

Findings

Derived bounds for the $q$-Mittag-Leffler function.

Applied bounds to solvability of fractional pseudo-parabolic equations.

Extended analysis to operators with discrete spectrum.

Abstract

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+\Gamma_q\left(1-\alpha\right)z}\leq e_{\alpha,1}\left(-z;q\right)\leq\frac{1}{1+\Gamma_q\left(\alpha+1\right)^{-1}z}. \end{eqnarray*} for all $0 < \alpha < 1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct and inverse problems for time-fractional pseudo-parabolic equations in quantum calculus for a large class of positive operators with discrete spectrum.