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Analysis of a Coupled System of \(\psi \)-Caputo Fractional Derivatives with Multipoint–Multistrip Integral Type Boundary Conditions

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Abstract

In this paper, we investigate the existence of solutions for a new coupled system of fractional differential equations that involves \(\psi \)-Caputo fractional derivatives equipped with coupled integro multistrip–multipoint boundary conditions. The uniqueness result for the given problem is obtained by utilizing the Banach contraction principle, while the existence results are established with the help of Schaefer’s fixed point theorem under specific assumptions. We also discuss the Ulam–Hyers stability for the problem at hand. Numerical examples are constructed for the illustration of the abstract results.

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References

  1. Abdo, M.S.: Qualitative analyses of \(\psi \)-Caputo type fractional integrodifferential equations in Banach spaces. J. Adv. Appl. Comput. Math. 2022, 9 (2022)

    Google Scholar 

  2. Abbas, N., Ali, M., Shatanawi, W., Mustafa, Z.: Thermodynamic properties of Second-grade micropolar nanofluid flow past an exponential curved Riga stretching surface with Cattaneo-Christov double diffusion. Alex. Eng. J. 81, 101–117 (2023)

    Google Scholar 

  3. Ahmad, M., Jiang, J., Zada, A., Ali, Z., Fu, Z., Xu, J.: Hyers–Ulam–Mittag–Leffler Stability for a system of fractional neutral differential equations. Discrete Dyn. Nat. Soc. 2020, 1–10 (2020)

    MathSciNet  Google Scholar 

  4. Ahmad, B., Kerthikeyan, P., Buvaneswari, K.: Fractional differential equations with coupled slit-strips type integral boundary conditions. AIMS Math. 4, 1596–1609 (2019)

    MathSciNet  Google Scholar 

  5. Ali, A., Khalid, S., Rahmat, G., Ali, G., Nisar, K.S., Alshahrani, B.: Controllability and Ulam–Hyers stability of fractional order linear systems with variable coefficients. Alex. Eng. J. 61(8), 6071–6076 (2022)

    Google Scholar 

  6. Ahmad, B., Ntouyas, S.K.: A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips type integral boundary conditions. J. Math. Sci. 226, 175–196 (2017)

    MathSciNet  Google Scholar 

  7. Agarwal, R.P., Assolami, A., Alsaedi, A., Ahmad, B.: Existence results and Ulam–Hyers stability for a fully coupled system of nonlinear sequential hilfer fractional differential equations and integro-multistrip–multipoint boundary conditions. Qual. Theory Dyn. Syst. 21, 125–158 (2022)

    MathSciNet  Google Scholar 

  8. Almeida, R.: A caputo fractional derivative of a function with respect to another function. Common. Nonlinear Sci. Numer. Sumer. 44, 460–481 (2014)

    MathSciNet  Google Scholar 

  9. Ali, S.M., Shatanawi, W., Kassim, M.D., Abdo, M.S., Saleh, S.: Investigating a class of generalized Caputo-type fractional integro-differential equations. J. Funct. Spaces 2022, 9 (2022)

    MathSciNet  Google Scholar 

  10. Abbas, N., Shatanawi, W., Hasan, F., Mustafa, Z.: Thermal analysis of MHD Casson–Sutterby fluid flow over exponential stretching curved sheet. Case Stud. Therm. Eng. 52, 103760 (2023)

    Google Scholar 

  11. Ali Khan, H.N., Zada, A., Popa, I.L., Ben Moussa, S.: Impulsive coupled Langevin \(\psi \)-Caputo Fractional Problem with Slit-Strips-Generalized Type Boundary Conditions. Fractal Fact. 7(12), 837–877 (2023)

    Google Scholar 

  12. Bastos, N.: Fractional Calculus on Time Scales (2012)

  13. Bekri, Z., Erturk, V.S., Kumar, P., Govindaraj, V.: Some novel analysis of two different Caputo-type fractional-order boundary value problems. Results Nonlinear Anal. 5(3), 299–311 (2022)

    Google Scholar 

  14. Carvalho, A., Pinto, C.M.A.: A delay fractional order model for the co-infection of malaria and HIV/AIDS. Int. J. Dyn. Cont. 5, 168–186 (2017)

    MathSciNet  Google Scholar 

  15. Debnath, L.: A brief historical introduction to fractional calculus. Int. J. Math. Ed. Sci. Technol. 35, 487–501 (2004)

    MathSciNet  Google Scholar 

  16. Ding, Y., Wang, Z., Ye, H.: Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Cont. Syst. Technol. 20, 763–769 (2012)

    Google Scholar 

  17. Faieghi, M., Kuntanapreeda, S., Delavari, H., et al.: LMI-based stabilization of a class of fractional order chaotic systems. Nonlinear Dyn. 72, 301–309 (2013)

    MathSciNet  Google Scholar 

  18. Govindaraj, V., Raju, K.G.: Controllability of fractional dynamical systems. A functional analytic approach. MCRF 7(4), 537–562 (2017)

    MathSciNet  Google Scholar 

  19. Govindaraj, V., Raju, K.G.: Functional approach to observability and controllability of linear fractional dynamical systems. JDSGT 15(2), 111–129 (2017)

    MathSciNet  Google Scholar 

  20. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Google Scholar 

  21. Javidi, M., Ahmad, B.: Dynamic analysis of time fractional order phytoplankton–touic phytoplankton–zooplankton system. Ecol. Model. 318, 8–18 (2015)

    Google Scholar 

  22. Jena, R.M., Chakraverty, S., Nisar, K.S.: Dynamical behavior of rotavirus epidemic model with non-probabilistic uncertainty under Caputo–Fabrizio derivative. Math. Methods Appl. Sci. 46(9), 10672–10697 (2023)

    MathSciNet  Google Scholar 

  23. Jadhav, C.P., Dale, T.B., Chinchane, V.L.: On Dirichlet problem of time-fractional advection–diffusion equation. J. Fract. Calc. Nonlinear Syst. 4(2), 1–13 (2023)

    Google Scholar 

  24. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order, methods, results and problem. Appl. Anal. 78, 153–192 (2013)

    MathSciNet  Google Scholar 

  25. Lv, Z., Ahmad, I., Xu, J., Zada, A.: Analysis of a hybrid coupled system Of \(\psi \)-Caputo fractional dervatives with generalized slit-strips type integral boundary conditions and impulses. Discrete Dyn. Nat. Soc. 6, 618–669 (2020)

    Google Scholar 

  26. Lundqvist, M.: Silicon Strip Detectors for Scanned Multi-slit u-Ray Imaging. Kungl Tekniska Hogskolan, Stockholm (2003)

    Google Scholar 

  27. Mellow, T., Karkkainen, L.: On the sound fields of infinitely long strips. J. Acoust. Soc. Am. 130, 153–167 (2011)

    Google Scholar 

  28. Morsy, A., Nisar, K.S., Ravichandran, C., Anusha, C.: Sequential fractional order neutral functional integro differential equations on time scales with Caputo fractional operator over Banach spaces. AIMS Math. 8(3), 5934–5949 (2023)

    MathSciNet  Google Scholar 

  29. Nisar, K.S., Anusha, C., Ravichandran, C.: A non-linear fractional neutral dynamic equations: existence and stability results on time scales. AIMS Math. 9(1), 1911–1925 (2024)

    MathSciNet  Google Scholar 

  30. Nisar, K.S., Munusamy, K., Ravichandran, C.: Results on existence of solutions in nonlocal partial functional integrodifferential equations with finite delay in nondense domain. Alex. Eng. J. 73, 377–384 (2023)

    Google Scholar 

  31. Poovarasan, R., Kumar, P., Nisar, K.S., Govindaraj, V.: The existence, uniqueness, and stability analyses of the generalized Caputo-type fractional boundary value problems. AIMS Math. 8(7), 1–28 (2023)

    MathSciNet  Google Scholar 

  32. Poovarasan, R., Kumar, P., Govindaraj, V., Murillo-Arcila, M.: The existence, uniqueness, and stability results for a nonlinear coupled system using \(\psi \)-Caputo fractional derivatives. Bound. Value Probl. 75, 2023 (2023)

    MathSciNet  Google Scholar 

  33. Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math. 26, 103–107 (2010)

    MathSciNet  Google Scholar 

  34. Rizwan, R., Zada, A., Wang, U.: Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. Adv. Differ. Equ. 85, 1–31 (2019)

    MathSciNet  Google Scholar 

  35. Sinan, M., Ansari, K.J., Kanwal, A., Shah, K., Abdeljawad, T., Abdalla, B.: Analysis of the mathematical model of cutaneous Leishmaniasis disease. Alex. Eng. J. 72, 117–134 (2023)

    Google Scholar 

  36. Vijayaraj, V., Ravichandran, C., Nisar, K.S., Valliammal, N., Logeswari, K., Albalawi, W., Abdel-Aty, A.: An outlook on the controllability of non-instantaneous impulsive neutral fractional nonlocal systems via Atangana–Baleanu–Caputo derivative. Arab J. Basic Appl. Sci. 30(1), 440–451 (2023)

    Google Scholar 

  37. Wang, G., Ahmad, B., Zhang, L.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74, 792–804 (2011)

    MathSciNet  Google Scholar 

  38. Wang, J., Zada, A., Li, W.: Ulams-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces. Int. J. Nonlinear Sci. Num. Sim. 19, 553–560 (2018)

    MathSciNet  Google Scholar 

  39. Yadav, P., Jahan, S., Shah, K., Peter, O.J., Abdeljawad, T.: Fractional-order modelling and analysis of diabetes mellitus: utilizing the Atangana–Baleanu Caputo (ABC) operator. Alex. Eng. J. 81, 200–209 (2023)

    Google Scholar 

  40. Yan, R., Sun, S., Lu, H., Zhao, Y.: Existence of solutions for fractional differential equations with integral boundary condition. Adv. Differ. Equ. 2014, 25–38 (2014)

    MathSciNet  Google Scholar 

  41. Zada, A., Ali, W., Farina, S.: Ulam–Hyers stability of nonlinear differential equations with fractional integrable impulsis. Math. Methods Appl. Sci. 40, 5502–5514 (2017)

    MathSciNet  Google Scholar 

  42. Zada, A., Ali, S., Li, Y.: Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Differ. Equ. 2017, 1–26 (2017)

    MathSciNet  Google Scholar 

  43. Zeidler, E.: Nonlinear Functional Analysis and Its Applications: II/B: Nonlinear Monotone Operators. Springer, Berlin (2013)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Haroon Niaz Ali Khan, Akbar Zada & Ishfaq Khan

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  1. Haroon Niaz Ali Khan
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  2. Akbar Zada
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  3. Ishfaq Khan
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Correspondence to Akbar Zada.

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Khan, H.N.A., Zada, A. & Khan, I. Analysis of a Coupled System of \(\psi \)-Caputo Fractional Derivatives with Multipoint–Multistrip Integral Type Boundary Conditions. Qual. Theory Dyn. Syst. 23, 129 (2024). https://doi.org/10.1007/s12346-024-00987-0

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  • DOI: https://doi.org/10.1007/s12346-024-00987-0

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