Research article

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

  • 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.

    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

    Related Papers:

  • 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.



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    [1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of the fractional differential equations, Vol. 204, North-Holland Mathematics Studies, Elsevier, 2006.
    [2] K. S. Miller, B. Ross, An introduction to the fractional calculus and differential equations, New York: Wiley, 1993.
    [3] I. Podlubny, Fractional differential equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Vol. 198, New York: Accademic Press, 1999.
    [4] M. I. Abbas, M. A. Ragusa, On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function, Symmetry, 13 (2021), 264. https://doi.org/10.3390/sym13020264 doi: 10.3390/sym13020264
    [5] D. A. Kattan, H. A. Hammad, Existence and stability results for piecewise Caputo-Fabrizio fractional differential equations with mixed delays, Fractal Fract., 7 (2023), 644. https://doi.org/10.3390/fractalfract7090644 doi: 10.3390/fractalfract7090644
    [6] S. Rezapour, S. K. Ntouyas, M. Q. Iqbal, A. Hussain, S. Etemad, J. Tariboon, An analytical survey on the solutions of the generalized double-order $\varphi$-integrodifferential equation, J. Funct. Spaces, 2021 (2021), 6667757. https://doi.org/10.1155/2021/6667757 doi: 10.1155/2021/6667757
    [7] A. Khan, K. Shah, T. Abdeljawad, M. A. Alqudah, Existence of results and computational analysis of a fractional order two strain epidemic model, Results Phys., 39 (2022), 105649. https://doi.org/10.1016/j.rinp.2022.105649 doi: 10.1016/j.rinp.2022.105649
    [8] S. Ayadi, O. Ege, M. De la Sen, On a coupled system of generalized hybrid pantograph equations involving fractional deformable derivatives, AIMS Math., 8 (2023), 10978–10996. https://doi.org/10.3934/math.2023556 doi: 10.3934/math.2023556
    [9] S. Ben Chikh, A. Amara, S. Etemad, S. Rezapour, On Hyers-Ulam stability of a multi-order boundary value problems via Riemann-Liouville derivatives and integrals, Adv. Differ. Equ., 2020 (2020), 547. https://doi.org/10.1186/s13662-020-03012-1 doi: 10.1186/s13662-020-03012-1
    [10] S. Etemad, M. M. Matar, M. A. Ragusa, S. Rezapour, Tripled fixed points and existence study to a tripled impulsive fractional differential system via measures of noncompactness, Mathematics, 10 (2022), 25. https://doi.org/10.3390/math10010025 doi: 10.3390/math10010025
    [11] O. Tunç, C. Tunç, Solution estimates to Caputo proportional fractional derivative delay integro-differential equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117 (2023), 12. https://doi.org/10.1007/s13398-022-01345-y doi: 10.1007/s13398-022-01345-y
    [12] S. Uçar, N. Özdemir, İ. Koca, E. Altun, Novel analysis of the fractional glucose-insulin regulatory system with non-singular kernel derivative, Eur. Phys. J. Plus., 135 (2020), 414. https://doi.org/10.1140/epjp/s13360-020-00420-w doi: 10.1140/epjp/s13360-020-00420-w
    [13] E. Uçar, N. Özdemir, A fractional model of cancer-immune system with Caputo and Caputo-Fabrizio derivatives, Eur. Phys. J. Plus., 136 (2021), 43. https://doi.org/10.1140/epjp/s13360-020-00966-9 doi: 10.1140/epjp/s13360-020-00966-9
    [14] N. Özdemir, S. Uçar, B. B. İ. Eroǧlu, Dynamical analysis of fractional order model for computer virus propagation with kill signals, Int. J. Nonlinear Sci. Numer. Simul., 21 (2020), 239–247. https://doi.org/10.1515/ijnsns-2019-0063 doi: 10.1515/ijnsns-2019-0063
    [15] S. Uçar, Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives, Discr. Contin. Dyn. Syst. Ser. S., 14 (2021), 2571–2589. https://doi.org/10.3934/dcdss.2020178 doi: 10.3934/dcdss.2020178
    [16] N. D. Phuong, F. M. Sakar, S. Etemad, S. Rezapour, A novel fractional structure of a multi-order quantum multi-integro-differential problem, Adv. Differ. Equ., 2020 (2020), 633. https://doi.org/10.1186/s13662-020-03092-z doi: 10.1186/s13662-020-03092-z
    [17] D. Baleanu, S. Rezapour, S. Etemad, A. Alsaedi, On a time-fractional equation via three-point boundary value conditions, Math. Probl. Eng., 2015 (2015), 785738. https://doi.org/10.1155/2015/785738 doi: 10.1155/2015/785738
    [18] F. H. Jackson, On $q$-definite integrals, Quart. J. Pure. Appl. Math., 41 (1910), 193–203.
    [19] F. H. Jackson, $q$-difference equations, Amer. J. Math., 32 (1910), 305–314. https://doi.org/10.2307/2370183
    [20] V. Fock, Zur theorie des Wasserstoffatoms, Z. Physik., 98 (1935), 145–154. https://doi.org/10.1007/BF01336904 doi: 10.1007/BF01336904
    [21] B. Ahmad, A. Alsaedi, S. K. Ntouyas, A study of second-order $q$-difference equations with boundary conditions, Adv. Differ. Equ., 2012 (2012), 35. https://doi.org/10.1186/1687-1847-2012-35 doi: 10.1186/1687-1847-2012-35
    [22] A. Boutiara, M. Benbachir, M. K. A. Kaabar, F. Martinez, M. E. Samei, M. Kaplan, Explicit iteration and unbounded solutions for fractional $q$-difference equations with boundary conditions on an infinite interval, J. Inequal. Appl., 2022 (2022), 29. https://doi.org/10.1186/s13660-022-02764-6 doi: 10.1186/s13660-022-02764-6
    [23] M. Houas, M. E. Samei, Existence and stability of solutions for linear and nonlinear damping of $q$-fractional Duffing-Rayleigh problem, Mediterr. J. Math., 20 (2023), 148. https://doi.org/10.1007/s00009-023-02355-9 doi: 10.1007/s00009-023-02355-9
    [24] S. Rezapour, A. Imran, A. Hussain, F. Martinez, S. Etemad, M. K. A. Kaabar, Condensing functions and approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs, Symmetry, 13 (2021), 469. https://doi.org/10.3390/sym13030469 doi: 10.3390/sym13030469
    [25] F. Wannalookkhee, K. Nonlaopon, M. Z. Sarikaya, H. Budak, M. A. Ali, On some new quantum trapezoid-type inequalities for $q$-differentiable coordinated convex functions, J. Inequal. Appl., 2023 (2023), 5. https://doi.org/10.1186/s13660-023-02917-1 doi: 10.1186/s13660-023-02917-1
    [26] J. Alzabut, M. Houas, M. I. Abbas, Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional $q$-derivatives, Demonstratio Math., 56 (2023), 20220205. https://doi.org/10.1515/dema-2022-0205 doi: 10.1515/dema-2022-0205
    [27] S. I. Butt, H. Budak, K. Nonlaopon, New variants of quantum midpoint-type inequalities, Symmetry, 14 (2022), 2599. https://doi.org/10.3390/sym14122599 doi: 10.3390/sym14122599
    [28] R. Chakrabarti, R. Jagannathan, A $(p, q)$-oscillator realization of two-parameter quantum algebras, J. Phys. A, 24 (1991), 5683–5701. https://doi.org/10.1088/0305-4470/24/13/002 doi: 10.1088/0305-4470/24/13/002
    [29] M. Mursaleen, M. Nasiruzzaman, A. Khan, K. J. Ansari, Some approximation results on Bleimann-Butzer-Hahn operators defined by $(p, q)$-integers, Filomat, 30 (2016), 639–648. https://doi.org/10.2298/FIL1603639M doi: 10.2298/FIL1603639M
    [30] M. Mursaleen, K. J. Ansari, A. Khan, Some approximation results by $(p, q)$-analogue of Bernstein-Stancu operators, Appl. Math. Comput., 264 (2015), 392–402. https://doi.org/10.1016/j.amc.2015.03.135 doi: 10.1016/j.amc.2015.03.135
    [31] K. Khan, D. K. Lobiyal, Bèzier curves based on Lupaş $(p, q)$-analogue of Bernstein functions in CAGD, J. Comput. Appl. Math., 317 (2017), 458–477. https://doi.org/10.1016/j.cam.2016.12.016 doi: 10.1016/j.cam.2016.12.016
    [32] A. Khan, V. Sharma, Statistical approximation by $(p, q)$-analogue of Bernstein-Stancu operators, Azerb. J. Math., 8 (2018), 100–121.
    [33] H. B. Jebreen, M. Mursaleen, M. Ahasan, On the convergence of Lupaş $(p, q)$-Bernstein operators via contraction principle, J. Inequal. Appl., 2019 (2019), 34. https://doi.org/10.1186/s13660-019-1985-y doi: 10.1186/s13660-019-1985-y
    [34] P. Njionou Sadjang, On the fundamental theorem of $(p, q)$-calculus and some $(p, q)$-Taylor formulas, Results Math., 73 (2018), 39. https://doi.org/10.1007/s00025-018-0783-z doi: 10.1007/s00025-018-0783-z
    [35] W. T. Cheng, W. H. Zhang, Q. B. Cai, $(p, q)$-gamma operators which preserve $x^2$, J. Inequal. Appl., 2019 (2019), 108. https://doi.org/10.1186/s13660-019-2053-3 doi: 10.1186/s13660-019-2053-3
    [36] G. V. Milovanović, V. Gupta, N. Malik, $(p, q)$-beta functions and applications in approximation, Bol. Soc. Mat. Mex., 24 (2018), 219–237. https://doi.org/10.1007/s40590-016-0139-1 doi: 10.1007/s40590-016-0139-1
    [37] J. Soontharanon, T. Sitthiwirattham, On fractional $(p, q)$-calculus, Adv. Differ. Equ., 2020 (2020), 35. https://doi.org/10.1186/s13662-020-2512-7
    [38] J. Soontharanon, T. Sitthiwirattham, Existence results of nonlocal Robin boundary value problems for fractional $(p, q)$-integrodifference equations, Adv. Differ. Equ., 2020 (2020), 342. https://doi.org/10.1186/s13662-020-02806-7 doi: 10.1186/s13662-020-02806-7
    [39] P. Neang, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, B. Ahmad, Nonlocal boundary value problems of nonlinear fractional $(p, q)$-difference equations, Fractal Fract., 5 (2021), 270. https://doi.org/10.3390/fractalfract5040270 doi: 10.3390/fractalfract5040270
    [40] P. Neang, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, B. Ahmad, Existence and uniqueness results for fractional $(p, q)$-difference equations with separated boundary conditions, Mathematics, 10 (2022), 767. https://doi.org/10.3390/math10050767 doi: 10.3390/math10050767
    [41] J. Soontharanon, T. Sitthiwirattham, On sequential fractional Caputo $(p, q)$-integrodifference equations via three-point fractional Riemann-Liouville $(p, q)$-difference boundary condition, AIMS Math., 7 (2022), 704–722. https://doi.org/10.3934/math.2022044 doi: 10.3934/math.2022044
    [42] W. Luangboon, K. Nonlaopon, J. Tariboon, S. K. Ntouyas, H. Budak, On generalizations of some integral inequalities for preinvex functions via $(p, q)$-calculus, J. Inequal. Appl., 2022 (2022), 157. https://doi.org/10.1186/s13660-022-02896-9 doi: 10.1186/s13660-022-02896-9
    [43] T. Sitthiwirattham, M. A. Ali, H. Budak, S. Etemad, S. Rezapour, A new version of $(p, q)$-Hermite-Hadamard's midpoint and trapezoidal inequalities via special operators in $(p, q)$-calculus, Bound. Value Probl., 2022 (2022), 84. https://doi.org/10.1186/s13661-022-01665-3 doi: 10.1186/s13661-022-01665-3
    [44] Z. Qin, S. Sun, On a nonlinear fractional $(p, q)$-difference Schrödinger equation, J. Appl. Math. Comp., 68 (2022), 1685–1698. https://doi.org/10.1007/s12190-021-01586-x doi: 10.1007/s12190-021-01586-x
    [45] A. Boutiara, J. Alzabut, M. Ghaderi, S. Rezapour, On a coupled system of fractional $(p, q)$-differential equation with Lipschitzian matrix in generalized metric space, AIMS Math., 8 (2023), 1566–1591. https://doi.org/10.3934/math.2023079 doi: 10.3934/math.2023079
    [46] R. P. Agarwal, H. Al-Hutami, B. Ahmad, On solvability of fractional $(p, q)$-difference equations with $(p, q)$-difference anti-periodic boundary conditions, Mathematics, 10 (2022), 4419. https://doi.org/10.3390/math10234419 doi: 10.3390/math10234419
    [47] G. Infante, J. R. L. Webb, Loss of positivity in a nonlinear scalar heat equation, Nonlinear Diff. Equ. Appl., 13 (2006), 249–261. https://doi.org/10.1007/s00030-005-0039-y doi: 10.1007/s00030-005-0039-y
    [48] P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math., 1 (2007), 311–323. https://doi.org/10.2298/AADM0701311R doi: 10.2298/AADM0701311R
    [49] C. R. Adams, The general theory of a class of linear partial $q$-difference equations, Trans. Amer. Math. Soc., 26 (1924), 283–312. https://doi.org/10.2307/1989141 doi: 10.2307/1989141
    [50] J. R. Graef, L. Kong, Positive solutions for a class of higher order boundary value problems with fractional $q$-derivatives, Appl. Math. Comput., 218 (2012), 9682–9689. https://doi.org/10.1016/j.amc.2012.03.006 doi: 10.1016/j.amc.2012.03.006
    [51] R. A. C. Ferreira, Positive solutions for a class of boundary value problems with fractional $q$-differences, Comput. Math. Appl., 61 (2011), 367–373. https://doi.org/10.1016/j.camwa.2010.11.012 doi: 10.1016/j.camwa.2010.11.012
    [52] L. E. J. Brouwer, Über Abbildunng von Mannigfaltigkeiten, Math. Ann., 71 (1911), 97–115. https://doi.org/10.1007/BF01456812 doi: 10.1007/BF01456812
    [53] O. Tunç, C. Tunç, On Ulam stabilities of delay Hammerstein integral equation, Symmetry, 15 (2023), 1736. https://doi.org/10.3390/sym15091736 doi: 10.3390/sym15091736
    [54] O. Tunç, C. Tunç, Ulam stabilities of nonlinear iterative integro-differential equations, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117 (2023), 118. https://doi.org/10.1007/s13398-023-01450-6 doi: 10.1007/s13398-023-01450-6
    [55] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad, Sci. USA., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [56] T. M. Rassias, On the stability of the linear mapping in Banach Spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
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