An analytical method for q − fractional dynamical equations on time scales

In this paper, we introduce the delta q−Laplace transform on a time scale and investigate some of its properties. We discuss some important properties of fractional delta q−calculus. Then, based on these properties and the q−Laplace transform, we propose an analytical method for solving a class of linear Caputo delta fractional dynamic equations (q−FDEs). This method relies on transforming the corresponding equation into an integer-order linear delta q−dynamic equation (q−DE). In fact, this transformation removes certain terms from a solution of the considered linear Caputo delta q−FDE, resulting in residual terms that satisfy the linear delta q−DE. Several examples are provided to demonstrate the effectiveness and efficiency of the proposed method.