Analytical solutions of $ q $-fractional differential equations with proportional derivative

In this paper, we aim to propose a novel $ q $-fractional derivative in the Caputo sense included proportional derivative. To this end, we firstly introduced a new concept of proportional $ q $-derivative and discussed its properties in detail. Then, we add this definition in the concept of Caputo derivative to state a new type of dynamical system with $ q $-calculus. For analytically solving this system, $ q $-Laplace transform has been successfully applied to obtain the solutions. Indeed, the bivariate Mittag-Leffler function has an essential role in this regard. Two illustrative examples are also given in detail.