Stability analysis on the post-quantum structure of a boundary value problem: application on the new fractional $(p,q)$-thermostat system

In this paper, we discussed some qualitative properties of solutions to a thermostat system in the framework of a novel mathematical model designed by the new $(p,q)$-derivatives in fractional post-quantum calculus. We transformed the existing standard model into a new control thermostat system with the help of the Caputo-like $(p,q)$-derivatives. By the properties of the $(p,q)$-gamma function and applying the fractional Riemann-Liouville-like $(p,q)$-integral, we obtained the equivalent $(p,q)$-integral equation corresponding to the given Caputo-like post-quantum boundary value problem ( $(p,q)$-BOVP) of the thermostat system. To conduct an analysis on the existence of solutions to this $(p,q)$-system, some theorems were proved based on the fixed point methods and the stability analysis was done from the Ulam-Hyers point of view. In the applied examples, we used numerical data to simulate solutions of the Caputo-like $(p,q)$-BOVPs of the thermostat system with respect to different parameters. The effects of given parameters in the model will show the performance of the thermostat system.