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

Reny George; Sina Etemad; Fahad Sameer Alshammari; Reny George; Sina Etemad; Fahad Sameer Alshammari

doi:10.3934/math.2024042

AIMS Mathematics

2024, Volume 9, Issue 1: 818-846. doi: 10.3934/math.2024042

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Research article

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

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Received: 09 September 2023 Revised: 13 November 2023 Accepted: 24 November 2023 Published: 04 December 2023
MSC : 26A51, 39A10, 39A13, 39A70

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.
- $ (p, q) $-fractional calculus,
- $ (p, q) $-gamma function,
- thermostat mathematical model,
- stability,
- fixed point
Citation: Reny George, Sina Etemad, Fahad Sameer Alshammari. Stability analysis on the post-quantum structure of a boundary value problem: application on the new fractional $ (p, q) $-thermostat system[J]. AIMS Mathematics, 2024, 9(1): 818-846. doi: 10.3934/math.2024042

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Abstract

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.

References

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