Q-fractional proportional derivatives and Q-laplace transforms

The main goal of this thesis is to develop a new type of q-fractional operators generated from proportional q-differences. To achieve this goal, first the main aspects and tools related to the q and q-fractional calculi are presented. After then, the proportional q-derivative is discussed. The proportional q-fractional differences or derivatives are proposed and the solutions of certain types of q-difference equations embodied by the proportional fractional derivatives are shown in details utilizing the q-Laplace transforms.