Theoretical and computational results for implicit non-singular hybrid fractional differential equation subject to multi-terms non-local initial conditions

Volume 36, Issue 2, pp 207--217 https://dx.doi.org/10.22436/jmcs.036.02.05

Publication Date: July 15, 2024 Submission Date: December 16, 2023 Revision Date: December 29, 2023 Accteptance Date: May 29, 2024

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 42 Downloads
• 259 Views

Authors

Shafiullah - Department of Mathematics, ‎University of Malakand, ‎Dir(L)‎, ‎Khyber Pakhtunkhwa, ‎Pakistan. K. Shah - Department of Mathematics, ‎University of Malakand, ‎Dir(L)‎, ‎Khyber Pakhtunkhwa, ‎Pakistan. - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎Riyadh 11586, ‎Saudi Arabia. M. Sarwar - Department of Mathematics, ‎University of Malakand, ‎Dir(L)‎, ‎Khyber Pakhtunkhwa, ‎Pakistan. - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎Riyadh 11586, ‎Saudi Arabia. Th. Abdeljawad - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎Riyadh 11586, ‎Saudi Arabia.

Abstract

‎This research work is devoted to investigate a class of hybrid fractional differential equations with \(n+1\) terms initial conditions‎. ‎The aforesaid problem is considered under the Atangana-Baleanue-Caputo fractional order derivative‎. ‎Here it is remarkable that hybrid differential equations with linear perturbations have significant applications in modeling various dynamical problems‎. ‎Sufficient conditions are established for the existence and uniqueness of solution to the problem under investigation by using the Banach and Krasnoselsikii's fixed point theorems‎. ‎Since stability theory plays important role in establishing various numerical and optimizations results‎, ‎therefore‎, ‎Hyers-Ulam type stability results are deduced for the considered problems using the tools of nonlinear functional analysis‎. ‎Additionally‎, ‎a numerical method based on Euler procedure is established to study some approbation results for the proposed problem‎. ‎By a pertinent example‎, ‎we demonstrate our results‎. ‎Also some graphical illustrations for different fractional orders are given‎.

Share and Cite

Keywords

• Atangana-Baleanu Caputo derivative
• Hyers-Ulam stability
• fixed point theorem
• Euler numerical method

MSC

•  26A33
•  34A08
•  35A09

References

• [1] N. Adjimi, A. Boutiara, M. E. Samei, S. Etemad, S. Rezapour, M. K. Kaabar, On solutions of a hybrid generalized Caputo-type problem via the noncompactness measure in the generalized version of Darbo’s criterion, J. Inequal. Appl., 2023 (2023), 23 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] Z. Ahmad, F. Ali, M. A. Almuqrin, S. Murtaza, F. Hasin, N. Khan, A. U. Rahman, I. Khan, Dynamics of love affair of Romeo and Juliet through modern mathematical tools: A critical analysis via fractal-fractional differential operator, Fractals, 30 (2022), 13 pages
  • View Article
  • Google Scholar
  • MATH


• [3] Z. Ahmad, F. Ali, A. M. Alqahtani, N. Khan, I. Khan, Dynamics of cooperative reactions based on chemical kinetics with reaction speed: A comparative analysis with singular and nonsingular kernels, Fractals, 30 (2022), 22 pages
  • View Article
  • Google Scholar
  • MATH


• [4] Z. Ahmad, F. Ali, N. Khan, I. Khan, Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel, Chaos Solitons Fractals, 153 (2021), 20 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [5] Z. Ahmad, G. Bonanomi, D. di Serafino, F. Giannino, Transmission dynamics and sensitivity analysis of pine wilt disease with asymptomatic carriers via fractal-fractional differential operator of Mittag-Leffler kernel, Appl. Numer. Math., 185 (2023), 446–465
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] S. Ahmed, A. T. Azar, M. Abdel-Aty, H. Khan, J. Alzabut, A nonlinear system of hybrid fractional differential equations with application to fixed time sliding mode control for Leukemia therapy, Ain Shams Eng. J., 15 (2024), 1–13
  • View Article
  • Google Scholar


• [7] T. M. Atanackovi´c, S. Pilipovi´c, D. Zorica, Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract. Calc. Appl. Anal., 21 (2018), 29–44
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] A. Atangana, Application of fractional calculus to epidemiology, Fractional Dynamics, 2015 (2015), 174–190
  • View Article
  • Google Scholar


• [9] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769
  • View Article
  • Google Scholar


• [10] D. Baleanu, A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernal, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 444–462
  • View Article
  • Google Scholar


• [11] M. Benyouba, S. Gu¨ lyaz O¨ zyurt, On extremal solutions of weighted fractional hybrid differential equations, Filomat, 38 (2024), 2091–2107
  • View Article
  • Google Scholar


• [12] T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85–88
  • View Article
  • MathSciNet
  • Google Scholar


• [13] S. O. Fatunla, Numerical methods for initial value problems in ordinary differential equations, Academic Press, Boston, MA (1988)
  • View Article
  • Google Scholar


• [14] A. Fernandez, D. Baleanu, The mean value theorem and Taylor’s theorem for fractional derivatives with Mittag-Leffler kernel, Adv. Difference Equ., 2018 (2018), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] R. George, S. Etemad, F. S. Alshammari, Stability analysis on the post-quantum structure of a boundary value problem: application on the new fractional (p, q)-thermostat system, AIMS Math., 9 (2024), 818–846
  • View Article
  • MathSciNet
  • Google Scholar


• [16] J. F. G´omez, L. Torres, R. F. Escobar, Fractional Derivatives with Mittag-Leffler Kernel:Trends and Applications in Science and Engineering, Springer, Cham (2019)
  • View Article
  • Google Scholar


• [17] R. Gul, K. Shah, Z. A. Khan, F. Jarad, On a class of boundary value problems under ABC fractional derivative, Adv. Difference Equ., 2021 (2021), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., River Edge, NJ (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [19] J. Isenberg, The initial value problem in general relativity, Springer, Dordrecht (2014)
  • View Article
  • Google Scholar


• [20] M. Izadi, S. Y¨ uzbasi, K. J. Ansari, Application of Vieta-Lucas series to solve a class of multi-pantograph delay differential equations with singularity, Symmetry, 13 (2021), 18 pages
  • View Article
  • Google Scholar


• [21] M. Jamil, R. A. Khan, K. Shah, Existence theory to a class of boundary value problems of hybrid fractional sequential integro-differential equations, Bound. Value Probl., 2019 (2019), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] N. Khan, F. Ali, Z. Ahmad, S. Murtaza, A. H. Ganie, I. Khan, S. M. Eldin, A time fractional model of a Maxwell nanofluid through a channel flow with applications in grease, Sci. Rep., 13 (2023), 15 pages
  • View Article
  • Google Scholar


• [23] R. A. Khan, S. Gul, F. Jarad, H. Khan, Existence results for a general class of sequential hybrid fractional differential equations, Adv. Difference Equ., 2021 (2021), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] H. Khan, A. Khan, F. Jarad, A. Shah, Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos Solitons Fractals, 131 (2020), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] H. Khan, Y. Li, W. Chen, D. Baleanu, A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Bound. Value Probl., 2014 (2017), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)
  • View Article
  • Google Scholar


• [27] P. J. Mosterman, G. Biswas, A comprehensive methodology for building hybrid models of physical systems, Artificial Intelligence, 121 (2000), 171–209
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] S. Murtaza, P. Kumam, T. Sutthibutpong, P. Suttiarporn, T. Srisurat, Z. Ahmad, Fractal-fractional analysis and numerical simulation for the heat transfer of ZnO+Al2O3 +TiO2=DWbased ternary hybrid nanofluid, ZAMM Z. Angew. Math. Mech., 104 (2024), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [29] H. Najafi, A. Bensayah, B. Tellab, S. Etemad, S. K. Ntouyas, S. Rezapour, J. Tariboon, Approximate numerical algorithms and artificial neural networks for analyzing a fractal-fractional mathematical model, AIMS Math., 8 (2023), 28280–28307
  • View Article
  • MathSciNet
  • Google Scholar


• [30] V. V. Nemytskii, V. V. Stepanov, Qualitative theory of differential equations, Princeton University Press, Princeton, NJ (1960)
  • View Article
  • Google Scholar


• [31] T. M. Rassias, P. ˇSemrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325–338
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] S. Rezapour, M. I. Abbas, S. Etemad, N. M. Dien, On a multi-point p-Laplacian fractional differential equation with generalized fractional derivatives, Math. Methods Appl. Sci., 46 (2023), 8390–8407
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] K. Shah, M. Sher, T. Abdeljawad, Study of evolution problem under Mittag-Leffler type fractional order derivative, Alex. Eng. J., 59 (2020), 3945–3951
  • View Article
  • Google Scholar


• [34] T. D. Skill, B. A. Young, Embracing the hybrid model: Working at the intersections of virtual and physical learning spaces, New Dir. Teach. Learn., 2002 (2002), 23–32
  • View Article
  • Google Scholar


• [35] A. M. Stuart, A. R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Rev., 36 (1994), 226–257
  • View Article
  • MathSciNet
  • Google Scholar


• [36] H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231
  • View Article
  • Google Scholar
  • MATH


• [37] J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, A. F. Galhano, Some applications of fractional calculus in engineering, Math. Probl. Eng., 2010 (2010), 34 pages
  • View Article
  • Google Scholar


• [38] Q. Yang, D. Chen, T. Zhao, Y. Chen, Fractional calculus in image processing: a review, Fract. Calc. Appl. Anal., 19 (2016), 1222–1249
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH