A Qualitative Study for Two Discrete Fractional Delta Difference BVPs with Falling Functions: Application on the Temperature Control System

Modeling of different processes and phenomena in real-world is one of the most important fields of the mathematics in which qualitative dynamics of such systems are studied from mathematical point of view. In this paper, we discuss the qualitative properties of solutions of a temperature control system in the context of a mathematical model in fractional discrete calculus. We discretize our supposed control system with the help of two delta sum and difference operators in the sense of the Caputo and Riemann–Liouville. By the existing properties of the falling functions, we obtain the equivalent difference formula corresponding to the given discrete delta difference boundary value problems of temperature control system. To conduct an analysis on solutions of this fractional system, the existence results are investigated via fixed points and the stability bahaviors are proved from the Ulam–Hyers point of view. In two applied examples, we use numerical data to simulate solutions of such discrete fractional delta boundary value problems of temperature control system.