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.

On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus

“…where the q-Mittag-Leffler function e α,β ðλ k ðt − q 2 sÞ α q ; qÞ is given by (see [25] and [26], Section 7):…”