To investigate a class of the singular fractional integro-differential quantum equations with multi-step methods

The objective of this paper is to investigate, by applying the standard Caputo fractional $q$--derivative of order $\alpha$, the existence of solutions for the singular fractional $q$--integro-differential equation $\mathcal{D}_q^\alpha [k](t) = \Omega (t , k(t), k'(t), \mathcal{D}_q^\beta [k](t), \int_0^t f(r) k(r) \, {\mathrm d}r )$, under some boundary conditions where $\Omega(t, k_1, k_2, k_3, k_4)$ is singular at some point $0 \leq t\leq 1$, on a time scale $\mathbb{T}_{ t_0} = \{ t : t =t_0q^n\}\cup \{0\}$, for  $n\in \mathbb{N}$ where  $t_0 \in \mathbb{R}$ and $q \in (0,1)$. We consider the compact map and avail the Lebesgue dominated theorem for finding solutions of the addressed problem. Besides, we prove the main results in context of completely continuous functions. Our attention is concentrated on fractional multi-step methods of both implicit and explicit type, for which sufficient existence conditions are investigated. Lastly, we present some examples involving  graphs, tables and algorithms to illustrate the validity of our theoretical findings.