On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

Abstract

This paper is devoted to explicit and numerical solutions to linear fractional q -difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q -derivative in q -calculus. The approaches based on the reduction to Volterra q -integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q -difference equations. For simplicity, we give results involving fractional q -difference equations of real order a > 0 and given real numbers in q -calculus. Numerical treatment of fractional q -difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.