A Study of Caputo Sequential Fractional Differential Equations with Mixed Boundary Conditions

Year 2025, Volume: 8 Issue: 2, 56 - 70, 27.06.2025

Djamel-eddine Hettadj , Djourdem Habib

https://doi.org/10.32323/ujma.1653542

Cited By: 1

Abstract

In this paper, we investigate the existence of solutions for a sequential fractional differential equation involving Caputo-type derivative subject to mixed boundary conditions. The core results are derived by employing Krasnoselskii's fixed point theorem and the Leray-Schauder fixed point theorem. We end this study by two illustrative numerical examples, which validate the applicability of our obtained results.

Keywords

Boundary value problem , Caputo fractional derivative , Existence , Fixed point theorem , Sequential fractional derivative

References

• [1] R. Gorenflo, F. Mainardi, Fractional Calculus, In: Fractals and Fractional Calculus in Continuum Mechanics, Vienna, Springer, 1997.
• [2] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific Publishing Company, Singapore, 2011.
• [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, Elsevier, New York, 2006.
• [4] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Elsevier, Amsterdam, The Netherlands, 1998.
• [5] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [6] D. Guyomar, B. Ducharne, G. Sebald, D. Audiger, Fractional derivative operators for modeling the dynamic polarization behavior as a function of frequency and electric field amplitude, IEEE Trans. Ultrason. Ferroelectr. Freq. Control., 56(3) (2009), 437-443. https://doi.org/10.1109/TUFFC.2009.1062
• [7] R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12(3) (2009), 299-318.
• [8] C. Ionescu, A. Lopes, D. Copot, J. T. Machado, J. H. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141-159.
• [9] A. Kilbas, S. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Cal. Appl. Anal., 7(3) (2004), 297-321.
• [10] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59(5) (2010), 1586-1593. https://doi.org/10.1016/j.camwa.2009.08.039
• [11] A. Al Elaiw, M. Manigandan, M. Awadalla, K. Abuasbeh, Mönch’s fixed point theorem in investigating the existence of a solution to a system of sequential fractional differential equations, AIMS Math., 8(2) (2023), 2591-2610. https://doi.org/10.3934/math.2023134
• [12] M. Awadalla, K. Abuasbeh, M. Subramanian, M. Manigandan, On a system of ψ-Caputo hybrid fractional differential equations with Dirichlet boundary conditions, Mathematics, 10(10) (2022), Article ID 1681. https://doi.org/10.3390/math10101681
• [13] N. Bouteraa, S. Benaicha, H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Univ. J. Math. Appl., 1(1) (2018), 39-45. https://doi.org/10.32323/ujma.396363
• [14] M. Manigandan, M. Subramanian, T. N. Gopal, B. Unyong, Existence and stability results for a tripled system of the Caputo type with multi-point and integral boundary conditions, Fractal Fract., 6(6) (2022), Article ID 285. https://doi.org/10.3390/fractalfract6060285
• [15] M. Subramanian, M. Manigandan, T. N. Gopal, Fractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions, Discontinuity Nonlinearity Complexity, 9(3) (2020), 421-431. https://doi.org/10.5890/DNC.2020.09.006
• [16] B. Ahmad, J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 64(10) (2012), 3046-3052. https://doi.org/10.1016/j.camwa.2012.02.036
• [17] R. Almeida, Fractional differential equations with mixed boundary conditions, Bull. Malays. Math. Sci. Soc. 42 (2019), 1687-1697. https://doi.org/10. 1007/s40840-017-0569-6
• [18] M. Awadalla, M. Murugesan, S. Muthaiah, J. Alahmadi, Existence results for the system of fractional-order sequential integrodifferential equations via Liouville–Caputo sense, J. Math., 2024(1) (2024), Article ID 6889622, 28 pages. https://doi.org/10.1155/2024/6889622
• [19] M. Awadalla, M. Murugesan, M. Kannan, J. Alahmadi, F. Al-Adsani, Utilizing Schaefer’s fixed point theorem in nonlinear Caputo sequential fractional differential equation systems, AIMS Math., 9(6) (2024), 14130-14157. https://doi.org/10.3934/math.2024687
• [20] H. Fazli, J. J. Nieto, Nonlinear sequential fractional differential equations in partially ordered spaces, Filomat, 32(13) (2018), 4577-4586. https://doi.org/10.2298/FIL1813577F
• [21] S. Muthaiah, M. Murugesan, M. Awadalla, B. Unyong, R. H. Egami, Ulam-Hyers stability and existence results for a coupled sequential Hilfer-Hadamard-type integrodifferential system, AIMS Math., 9(6) (2024), 16203-16233. https://doi.org/10.3934/math.2024784
• [22] A. Salem, L. Almaghamsi, Solvability of sequential fractional differential equation at resonance, Mathematics, 11(4) (2023), Article ID 1044. https://doi.org/10.3390/math11041044
• [23] N. I. Mahmudov, M. Awadalla, K. Abuassba, Nonlinear sequential fractional differential equations with nonlocal boundary conditions, Adv. Differ. Equ., 2017 (2017), Article ID 319, 1-15. https://doi.org/10.1186/s13662-017-1371-3
• [24] M. Awadalla, K. Abuasbeh, M. Manigandan, A. A. Al Ghafli, H. J. Al Salman, Applicability of Darbo’s fixed point theorem on the existence of a solution to fractional differential equations of sequential type, J. Math., 2023(1) (2023), 1-19. https://doi.org/10.1155/2023/7111771
• [25] D. Yan, Boundary problems of sequential fractional differential equations having a monomial coefficient, Heliyon, 10(7) (2024), Article ID e36538, 1-14. https://doi.org/10.1016/j.heliyon.2024.e36538
• [26] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
• [27] M. A. Krasnoselskii, Two remarks on the method of successive approximations, Usp. Mat. Nauk, 10 (1955), 123-127.
• [28] A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York, 2003.