The Cauchy problems for q-difference equations with the Caputo fractional derivatives

Abstract

The fractional differential equations play important roles due to their numerous applications and also for the important role they play not only in mathematics but also in other sciences. In the present research work, we build up the explicit solutions to linear fractional q-differential equations with the q-Caputo fractional derivative of real order a > 0. To speak more precisely, we will achieve our main results we use that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. After that, by using the method of successive approximations is applied to solve the Volterra q-integral equation we construct the the explicit solutions to linear fractional q-differential equations. In the same way we have the more general homogeneous fractional q-differential equation with the Caputo fractional q-derivative of real order a > 0 and we give other The (Mittag-Leffler) q-function. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these explicit solutions.