On quantum trigonometric fractional calculus

Abstract

In this research work, we introduce an innovative concept known as the q-trigonometric derivative within the framework of the Caputo derivative. Our approach begins by introducing a novel notion of a trigonometric q-derivative and thoroughly examining its characteristics. We subsequently merge this new definition with the Caputo derivative to introduce a novel approach to q-fractional calculus. To analytically address this q-trigonometric system, we effectively employ the q-Laplace transform to derive solutions. Notably, the bivariate q-Mittag-Leffler (q-ML) function plays a significant role in this process. We provide detailed explanations and examples of this approach with two illustrative examples.

Keywords

• q-trigonometric
• Caputo trigonometric q-derivative
• Fractional derivatives
• Differential equations