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Existence and uniqueness of blow-up solution to a fully fractional thermostat model

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  • Saha, Kiran Kumar
  • Sukavanam, N.

Abstract

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

Suggested Citation

  • Saha, Kiran Kumar & Sukavanam, N., 2023. "Existence and uniqueness of blow-up solution to a fully fractional thermostat model," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:chsofr:v:174:y:2023:i:c:s0960077923007488
    DOI: 10.1016/j.chaos.2023.113847
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    References listed on IDEAS

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    1. Yong Liang, David I. Levine, and Zuo-Jun (Max) Shen, 2012. "Thermostats for the Smart Grid: Models, Benchmarks, and Insights," The Energy Journal, International Association for Energy Economics, vol. 0(Number 4).
    2. Haq, Abdul & Sukavanam, N., 2020. "Existence and approximate controllability of Riemann-Liouville fractional integrodifferential systems with damping," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Haq, Abdul & Sukavanam, N., 2022. "Existence and partial approximate controllability of nonlinear Riemann–Liouville fractional systems of higher order," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    4. Georgios D. Kontes & Georgios I. Giannakis & Philip Horn & Simone Steiger & Dimitrios V. Rovas, 2017. "Using Thermostats for Indoor Climate Control in Office Buildings: The Effect on Thermal Comfort," Energies, MDPI, vol. 10(9), pages 1-22, September.
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