On a q-Analogue of the Right Local General Truncated M-Fractional Derivative

Abstract

We introduce a q-analogue of the right local general truncated M-fractional derivative for α-differentiable functions. From this newly defined operator, q-analogues of the standard properties and results of the -right local general truncated M-fractional derivative like the Rolle’s theorem, the mean value theorem and its extension, inverse property, the fundamental theorem of calculus and the theorem of integration by parts are obtained. In context with this q-fractional derivative operator, a q-analogue of a physical problem, the falling body problem, is obtained. Also, the q-vertical velocity and the q-distance are obtained from this problem and the solutions has been compared and shown in the graphs for various combination of q-parameter and fractional order α with the classical ordinary
solution.

Key words and phrases. q-derivative; q-integral; q-calculus; Falling body problem.

2010 Mathematics Subject Classification. 05A30, 26A33, 33D05, 39A13