Skip to main content
SpringerLink
Log in

Modeling and analysis of an implicit fractional order differential equation with multiple first-order fractional derivatives and non-local boundary conditions

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

In recent years, researchers have investigated boundary value problems of single terms and, in rare circumstances, multi-term fractional differential equations. Nonetheless, differential equations containing more than one fractional differential operator of the implicate type must be formulated in some cases. Therefore, bearing in mind the significance of multi-term fractional order in the study of implicit differential equations, we propose a novel implicit kind of multi-terms fractional delay differential equations (MFDDEs) supplemented with non-local boundary conditions. In addition, using common fixed point theorems, conditions for the existence and uniqueness of the solution will be created. Moreover, investigations correlated to Ulam’s type of stability will also be the focus of our consideration. The significance of the newly formulated delay type of fractional order differential equation with implicit behavior and non-local boundary conditions will then be demonstrated using an example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. V.E. Tarasov (ed.), Applications in physics, part b (Walter de Gruyter GmbH & Co KG, 2019)

    MATH  Google Scholar 

  2. W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332(1), 709–26 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.F. Ali, M. Sharma, R. Jain, An application of fractional calculus in electrical engineering. Adv. Eng. Technol. Appl. 5(4), 41–5 (2016)

    Article  Google Scholar 

  4. R. Koeller, Applications of fractional calculus to the theory of visco elasticity. J. Appl. Mech. (1984). https://doi.org/10.1115/1.3167616

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Fallahgoul, S. Focardi, F. Fabozzi, Fractional calculus and fractional processes with applications to financial economics: theory and application (Academic Press, 2016)

    MATH  Google Scholar 

  6. H. Khan, J. Alzabut, H. Gulzar, Existence of solutions for hybrid modified ABC-fractional differential equations with p-Laplacian operator and an application to a waterborne disease model. Alex. Eng. J. 1(70), 665–72 (2023)

    Article  Google Scholar 

  7. H. Khan, J. Alzabut, O. Tunç, M.K. Kaabar, A fractal-fractional COVID-19 model with a negative impact of quarantine on the diabetic patients. Results Control Optim 1(10), 100199 (2023)

    Article  Google Scholar 

  8. A. Ali, K. Shah, Y. Li, R.A. Khan, Numerical treatment of coupled system of fractional order partial differential equations. J. Math. Comput. Sci. 19, 74–85 (2019)

    Article  Google Scholar 

  9. H. Khan, J. Alzabut, A. Shah, Z.Y. He, S. Etemad, S. Rezapour, A. Zada, On fractal-fractional waterborne disease model: a study on theoretical and numerical aspects of solutions via simulations. Fractals 26, 2340055 (2023)

    Article  MATH  Google Scholar 

  10. S. Hussain, E.N. Madi, H. Khan, M.S. Abdo, A numerical and analytical study of a stochastic epidemic SIR model in the light of white noise. Adv. Math. Phys. 27, 2022 (2022)

    MathSciNet  MATH  Google Scholar 

  11. H. Khan, F. Ahmad, O. Tunç, M. Idrees, On fractal-fractional Covid-19 mathematical model. Chaos Solit. Fractals. 1(157), 111937 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Alzabut, G. Alobaidi, S. Hussain, E.N. Madi, H. Khan, Stochastic dynamics of influenza infection: qualitative analysis and numerical results. Math. Biosci. Eng. 1(19), 10316–31 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Ali, K. Shah, D. Ahmad, G.U. Rahman, N. Mlaiki, T. Abdeljawad, Study of multi term delay fractional order impulsive differential equation using fixed point approach. AIMS Math. 7(7), 11551–80 (2022)

    Article  MathSciNet  Google Scholar 

  14. A. Shah, R. Ali Khan, H. Khan, A fractional-order hybrid system of differential equations: existence theory and numerical solutions. Math. Methods Appl. Sci. 45(7), 4024–34 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  15. A. Ali, K. Shah, Y. Li, Topological degree theory and Ulam’s stability analysis of a boundary value problem of fractional differential equations. Frontiers in functional equations and analytic inequalities (Springer, Cham, 2019), pp.73–92

    MATH  Google Scholar 

  16. H. Khan, K. Alam, H. Gulzar, S. Etemad, S. Rezapour, A case study of fractal-fractional tuberculosis model in China: existence and stability theories along with numerical simulations. Math. Comput. Simul. 1(198), 455–73 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some in science and engineering, 1st edn. (Academic Press, Cham, 1998)

    Google Scholar 

  18. X. Liu, L. Liu, Y. Wu, Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives. Bound. Value Probl. 2018(1), 1–21 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Vu, N. Van Hoa, Hyers-Ulam stability of fuzzy fractional Volterra integral equations with the kernel \(\psi \)-function via successive approximation method. Fuzzy Sets Syst. 419, 67–98 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. K. Shah, W. Hussain, Investigating a class of nonlinear fractional differential equations and its Hyers-Ulam stability by means of topological degree theory. Numer. Funct. Anal. Optim. 40(12), 1355–72 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. V.S. Ertürk, A. Ali, K. Shah, P. Kumar, T. Abdeljawad, Existence and stability results for nonlocal boundary value problems of fractional order. Bound. Value Probl. 2022(1), 1–5 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Boutiara, M.M. Matar, J. Alzabut, M.E. Samei, H. Khan, On ABC coupled Langevin fractional differential equations constrained by Perov’s fixed point in generalized Banach spaces. AIMS Math. 8(5), 12109–32 (2023)

    Article  MathSciNet  Google Scholar 

  23. A. Zada, S. Shaleena, T. Li, Stability analysis of higher order nonlinear differential equations in \(\beta \)-normed spaces. Math. Methods Appl. Sci. 42, 1151–1166 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. A. Zada, W. Ali, S. Farina, Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. 40(15), 5502–5514 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. A. Zada, P. Wang, D. Lassoued, T. Li, Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear non-autonomous systems. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1248-5

    Article  MATH  Google Scholar 

  26. K. Shah, A. Ullah, J.J. Nieto, Study of fractional order impulsive evolution problem under nonlocal Cauchy conditions. Math. Methods Appl. Sci. 44(11), 8516–8527 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Sher, K. Shah, J. Rassias, On qualitative theory of fractional order delay evolution equation via the prior estimate method. Math. Methods Appl. Sci. 43(10), 6464–6475 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. M. Benchohra, J.E. Lazreg, Existence results for nonlinear implicit fractional differential equations. Surv. Math. Appl. 9, 79–92 (2014)

    MathSciNet  MATH  Google Scholar 

  29. M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order. Moroc. J. Pure Appl. Anal. 1(1), 1–6 (2015)

    Article  MATH  Google Scholar 

  30. J.R. Ockendon, A.B. Tayler, The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. A. 322(1551), 447–68 (1971)

    Article  ADS  Google Scholar 

  31. T. Müller, M. Lauk, M. Reinhard et al., Estimation of delay times in biological systems. Ann. Biomed. Eng. 31(11), 1423–1439 (2003)

    Article  Google Scholar 

  32. M.R. Roussel, The use of delay differential equations in chemical kinetics. J. Phys. Chem. 100(20), 8323–8330 (1996)

    Article  Google Scholar 

  33. E. Bayraktar, M. Egami, The effects of implementation delay on decision-making under uncertainty. Stoch. Process. Appl. 117, 333–358 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. G. ur Rahman, R.P. Agarwal, D. Ahmad, Existence and stability analysis of nth order multi term fractional delay differential equation. Chaos Solit. Fractals. 155, 111709 (2022)

    Article  MATH  Google Scholar 

  35. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations (Elsevier, 2006)

    MATH  Google Scholar 

  36. D. Baleanu, S. Etemad, H. Mohammadi, S. Rezapour, A novel modeling of boundary value problems on the glucose graph. Commun. Nonlinear Sci. Numer. Simul. 100, 105844 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular Kernel. Prog. Fract. Differ. Appl. 1(2), 73–85 (2015)

    Google Scholar 

  38. D. Baleanu, M.S. Osman, A. Zubair, N. Raza, O.A. Arqub, W.X. Ma, Soliton solutions of a nonlinear fractional Sasa-Satsuma equation in monomode optical fibers. Appl. Math. Inform. Sci. 14(3), 365–74 (2020)

    Article  MathSciNet  Google Scholar 

  39. F.S. Bayones, K.S. Nisar, K.A. Khan, N. Raza, N.S. Hussien, M.S. Osman, K.M. Abualnaja, Magneto-hydrodynamics (MHD) flow analysis with mixed convection moves through a stretching surface. AIP Adv. 11(4), 04500 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Swat, District Swat, Charbagh, Pakistan

    Ghaus ur Rahman & Dildar Ahmad

  2. CONACyT-Tecnologico Nacional de Mexico CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, Morelos, Mexico

    J. F. Gómez-Aguilar

Authors
  1. Ghaus ur Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. F. Gómez-Aguilar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dildar Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

GuR and DA performed the validation, formal analysis, and investigation; JFG-A performed the conceptualization, methodology, validation, and writing-review and editing; GuR performed the writing-original draft preparation and writing review and editing. All authors have read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to J. F. Gómez-Aguilar.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rahman, G.u., Gómez-Aguilar, J.F. & Ahmad, D. Modeling and analysis of an implicit fractional order differential equation with multiple first-order fractional derivatives and non-local boundary conditions. Eur. Phys. J. Spec. Top. 232, 2367–2383 (2023). https://doi.org/10.1140/epjs/s11734-023-00961-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-023-00961-y

Search

Navigation