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. Keywords: q-calculus; q-proportional; proportional Caputo q-derivative; q-Laplace transform Mathematics Subject Classification: 26A33, 39A13