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MATHEMATICAL ANALYSIS OF COUPLED SYSTEMS WITH FRACTIONAL ORDER BOUNDARY CONDITIONS

ZEESHAN ALI

Department of Mathematics, University of Peshawar, Peshawar 2500, Pakistan

E-mail Address: zeeshanmaths1@gmail.com

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KAMAL SHAH

Department of Mathematics, University of Malakand, Dir(L) 18800, Pakistan

E-mail Address: kamalshah408@gmail.com

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AKBAR ZADA

Department of Mathematics, University of Peshawar, Peshawar 2500, Pakistan

E-mail Address: zadababo@yahoo.com

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, and

POOM KUMAM

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science King Mongkut’s University, of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

E-mail Address: poom.kumam@mail.kmutt.ac.th

E-mail Address: kumampoom@gmail.com

Corresponding author.

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https://doi.org/10.1142/S0218348X20400125Cited by:18 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal and Fractional with Applications to Nature
Guest Editors: Abdon Atangana, Emile Franc Goufo Ndongmo, José Francisco Gómez Aguilar, Muhammad Altaf Khan and Ilknur Koca

Abstract

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions for the stability mentioned above by two methods. Also, an example is given to illustrate our theoretical results. The proposed problem is the generalization of third-order ordinary differential equations with classical, initial and anti-periodic boundary conditions.

Keywords:

References

1. B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009) 1838–1843. Crossref, Web of Science, Google Scholar
2. B. Ahmad and S. K. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal. 15(3) (2012) 362–382. Crossref, Web of Science, Google Scholar
3. B. Ahmad, S. K. Ntouyas, A. Alsaed, W. Shammakh and R. P. Agarwal, Existence theory for fractional differential equations with non-separated type nonlocal multi-point and multi-strip boundary conditions, Adv. Differ. Equ. 89 (2018) 2018, https://doi.org/10.1186/s13662-018-1546-6. Google Scholar
4. A. R. Aftabizadeh, Y. K. Huang and N. H. Pavel, Nonlinear third-order differential equations with anti-periodic boundary conditions and some Optimal control problems, J. Math. Anal. Appl. 192 (1995) 266–293. Crossref, Web of Science, Google Scholar
5. R. P. Agarwal, A. Cabada, V. Otero-Espinar and S. Dontha, Existence and uniqueness of solutions for anti-periodic difference equations, Arch. Inequal. Appl. 2 (2004) 397–411. Google Scholar
6. R. P. Agarwal, B. Ahmad and A. Alsaedi, Fractional-order differential equations with anti-periodic boundary conditions: A survey, Bound. Value Probl. 173 (2017) 173. https://doi.org/10.1186/s13661-017-0902-x Crossref, Web of Science, Google Scholar
7. R. P. Agarwal, D. O’Regan and S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl Math. 60(6) (2015) 653–676. Crossref, Web of Science, Google Scholar
8. B. Ahmad and J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear Anal. 69 (2008) 3291–3298. Crossref, Web of Science, Google Scholar
9. Z. Ali, A. Zada and K. Shah, Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem, Bound. Value Probl. 2018 (2018) 175, https://doi.org/10.1186/s13661-018-1096-6. Crossref, Web of Science, Google Scholar
10. Z. Ali, P. Kumam, K. Shah and A. Zada, Investigation of Ulam stability results of a coupled system of nonlinear implicit fractional differential equations, Mathematics 7(4) (2019) 341, https://doi.org/10.3390/math7040341. Crossref, Web of Science, Google Scholar
11. M. Benchohra, S. Hamani and S. K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009) 2391–2396. Crossref, Web of Science, Google Scholar
12. C. Bai and J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150 (2004) 611–621. Web of Science, Google Scholar
13. Y. Chen and H. An, Numerical solutions of coupled Burgers equations with time and space fractional derivatives, Appl. Math. Comput. 200 (2008) 87–95. Crossref, Web of Science, Google Scholar
14. H. L. Chen, Antiperiodic wavelets, J. Comput. Math. 14 (1996) 32–39. Web of Science, Google Scholar
15. J. O. C. Ezeilo, A property of the phase space trajectories of a third order nonlinear ordinary differential equation, J. London Math. Soc. 37 (1962) 33–41. Crossref, Google Scholar
16. C. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl. 61 (2011) 191–202. Crossref, Web of Science, Google Scholar
17. A. Granas and J. Dugundji, Fixed Point Theory (Springer, New York, 2005). Google Scholar
18. C. P. Gupta, On a third-order three-point boundary value problem at resonance, Differ. Integral Equ. 2 (1989) 1–12. Google Scholar
19. M. Gregus, Third Order Linear Differential Equations (Reidel, Dordrecht, 1987). Crossref, Google Scholar
20. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000). Link, Google Scholar
21. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27(4) (1941) 222–224. Crossref, Google Scholar
22. H. Khan, J. F. Gómez-Aguilar, A. Khan and T. S. Khan, Stability analysis for fractional order advection-reaction diffusion system, Physica A 521 (2019) 737–751. Crossref, Web of Science, Google Scholar
23. D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables (Birkhäiuser, Boston, 1998). Crossref, Google Scholar
24. S. M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19 (2006) 854–858. Crossref, Web of Science, Google Scholar
25. A. Khan, K. Shah, Y. Li and T. S. Khan, Ulam type stability for a coupled systems of boundary value problems of nonlinear fractional differential equations, J. Funct. Spaces 2017 (2017) 8. Web of Science, Google Scholar
26. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204 (Elsevier, Amsterdam, 2006). Google Scholar
27. J. P. LaSalle, Stability theory for ordinary differential equations, J. Differ. Equ. 4(1) (1986) 57–65. Crossref, Web of Science, Google Scholar
28. F. Liu and K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl. 62 (2011) 822–833. Crossref, Web of Science, Google Scholar
29. F. Meral, T. Royston and R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 939–945. Crossref, Web of Science, Google Scholar
30. K. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw. 41 (2010) 9–12. Crossref, Web of Science, Google Scholar
31. I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26 (2010) 103–107. Web of Science, Google Scholar
32. T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. Crossref, Web of Science, Google Scholar
33. U. Riaz et al., Analysis of nonlinear coupled systems of impulsive fractional differential equations with Hadamard derivatives, Math. Probl. Eng. 2019 (2019) 5093572. Crossref, Web of Science, Google Scholar
34. J. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with timevarying delays, Phys. Lett. A 372 (2008) 5011–5016. Crossref, Web of Science, Google Scholar
35. S. Sherman, A third-order nonlinear system arising from a nuclear spin generator, Contrib. Diff. Equ. 2 (1963) 197–227. Google Scholar
36. T. C. Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differ. Equ. 259(9) (2015) 4897–4923. Crossref, Web of Science, Google Scholar
37. K. Shah and C. Tunç, Existence theory and stability analysis to a system of boundary value problem, J. Taibah Univ. Sci. 11 (2017) 1330–1342. Crossref, Web of Science, Google Scholar
38. J. Wang, K. Shah and A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods. Appl. Sci. 41 (2018) 2392–2402. Crossref, Web of Science, Google Scholar
39. S. M. Ulam, A Collection of the Mathematical Problems (Interscience, New York, 1960). Google Scholar
40. J. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett. 73 (2017) 157–162. Crossref, Web of Science, Google Scholar
41. J. Wang, M. Fečkan and Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bull. Sci. Math. 141 (2017) 727–746. Crossref, Web of Science, Google Scholar
42. L. L. Rauch, Oscillation of a third order nonlinear autonomous system, in Contributions to the Theory of Nonlinear Oscillations, Vol. 1, ed. S. Lefschetz (Princeton University Press, 1950), pp. 39–88. Crossref, Google Scholar
43. A. Zada, S. Fatima, Z. Ali, J. Xu and Y. Cui, Stability results for a coupled system of impulsive fractional differential equations, Mathematics 7 (2019) 927. Crossref, Web of Science, Google Scholar

Vol. 28, No. 08

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History

Received 24 December 2019

Accepted 27 February 2020

Published: July 10, 2020

Information

This is an Open Access article in the “Special Issue on Fractal and Fractional with Applications to Nature” published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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MATHEMATICAL ANALYSIS OF COUPLED SYSTEMS WITH FRACTIONAL ORDER BOUNDARY CONDITIONS

Abstract

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