Investigation of 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 a, the existence of solutions for a class of the singular fractional q - integro-differential equation under some boundary conditions on a time scale. We consider the compact map and the Lebesgue dominated theorem for nding solu- tions of the problem. 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 ndings.