Existence and analysis of Hilfer-Hadamard fractional differential equations in RLC circuit models

Murugesan Manigandan; R. Meganathan; R. Sathiya Shanthi; Mohamed Rhaima; Murugesan Manigandan; R. Meganathan; R. Sathiya Shanthi; Mohamed Rhaima

doi:10.3934/math.20241394

AIMS Mathematics

2024, Volume 9, Issue 10: 28741-28764. doi: 10.3934/math.20241394

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Existence and analysis of Hilfer-Hadamard fractional differential equations in RLC circuit models

1.
Center for Nonlinear and Complex Networks, SRM TRP Engineering College, Tiruchirappalli 621 105, Tamil Nadu, India
2.
Center for Research, Easwari Engineering College, Chennai 600 089, Tamil Nadu, India
3.
Department of Mathematics, School of Engineering and Technology, Dhanalakshmi Srinivasan University Trichy 621112, Tamil Nadu, India
4.
Department of Mathematics, School of Agricultural Sciences, Dhanalakshmi Srinivasan University, Trichy 621112, Tamil Nadu, India
5.
Department of Statistics and Operations Research, College of Sciences, King Saud University, P.O. Box 2455 Riyadh 11451, Saudi Arabia

Received: 11 July 2024 Revised: 05 September 2024 Accepted: 30 September 2024 Published: 11 October 2024
MSC : 34A05, 34B15, 47H10

This paper explores a fractional integro-differential equation with boundary conditions that incorporate the Hilfer-Hadamard fractional derivative. We model the RLC circuit using fractional calculus and define weighted spaces of continuous functions. The existence and uniqueness of solutions are established, along with their Ulam-Hyers and Ulam-Hyers-Rassias stability. Our analysis employs Schaefer's fixed-point theorem and Banach's contraction principle. An illustrative example is presented to validate our findings.
- fractional differential equations,
- Hilfer-Hadamard fractional derivative,
- non-local boundary conditions,
- existence,
- fixed-point theorem
Citation: Murugesan Manigandan, R. Meganathan, R. Sathiya Shanthi, Mohamed Rhaima. Existence and analysis of Hilfer-Hadamard fractional differential equations in RLC circuit models[J]. AIMS Mathematics, 2024, 9(10): 28741-28764. doi: 10.3934/math.20241394

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Abstract

This paper explores a fractional integro-differential equation with boundary conditions that incorporate the Hilfer-Hadamard fractional derivative. We model the RLC circuit using fractional calculus and define weighted spaces of continuous functions. The existence and uniqueness of solutions are established, along with their Ulam-Hyers and Ulam-Hyers-Rassias stability. Our analysis employs Schaefer's fixed-point theorem and Banach's contraction principle. An illustrative example is presented to validate our findings.

References

[1]	S. Abbas, M. Benchohra, J. E. Lagreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Differ. Equ., 2017 (2017), 180. https://doi.org/10.1186/s13662-017-1231-1 doi: 10.1186/s13662-017-1231-1
[2]	S. Abbas, M. Benchohra, J. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability, Chaos Soliton. Fract., 102 (2017), 47–71. https://doi.org/10.1016/j.chaos.2017.03.010 doi: 10.1016/j.chaos.2017.03.010
[3]	B. Ahmad, S. K Ntouyas, Hilfer-Hadamard fractional boundary value problems with nonlocal mixed boundary conditions, Fractal Fract., 5 (2021), 195. https://doi.org/10.3390/fractalfract5040195 doi: 10.3390/fractalfract5040195
[4]	M. Subramanian, J. Alzabut, D. Baleanu, M. E. Samei, A. Zada, Existence, uniqueness and stability analysis of a coupled fractional-order differential systems involving Hadamard derivatives and associated with multi-point boundary conditions, Adv. Differ. Equ., 2021 (2021), 267. https://doi.org/10.1186/s13662-021-03414-9 doi: 10.1186/s13662-021-03414-9
[5]	D. Baleanu, R. P. Agarwal, R. K. Parmar, M. Alqurashi, S. Salahshour, Extension of the fractional derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10 (2017), 2914–2924. https://doi.org/10.22436/jnsa.010.06 doi: 10.22436/jnsa.010.06
[6]	Y. Başcı, S. Öğrekçi, A. Mısır, On Hyers-Ulam stability for fractional differential equations including the new Caputo-Fabrizio fractional derivative, Mediterr. J. Math., 16 (2019), 131. https://doi.org/10.1007/s00009-019-1407-x doi: 10.1007/s00009-019-1407-x
[7]	P. Borisut, P. Kumam, I. Ahmed, K. Sitthithakerngkiet, Nonlinear Caputo fractional derivative with nonlocal Riemann-Liouville fractional integral condition via fixed-point theorems, Symmetry, 11 (2019), 829. https://doi.org/10.3390/sym11060829 doi: 10.3390/sym11060829
[8]	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 (2022), 25. https://doi.org/10.1186/s13661-022-01606-0 doi: 10.1186/s13661-022-01606-0
[9]	J. Hadamard, Essai sur l'etude des fonctions donnes par leur developpement de Taylor, J. Math. Pure. Appl., 8 (1892), 101–186.
[10]	E. Fadhal, K. Abuasbeh, M. Manigandan, M. Awadalla, Applicability of Mönch's fixed-point theorem on a system of (k, $\psi$)-Hilfer type fractional differential equations, Symmetry, 14 (2022), 2572. https://doi.org/10.3390/sym14122572 doi: 10.3390/sym14122572
[11]	H. Fan, J. Tang, K. Shi, Y. Zhao, Hybrid impulsive feedback control for drive-response synchronization of fractional-order multi-link memristive neural networks with multi-delays, Fractal Fract., 7 (2023), 495. https://doi.org/10.3390/fractalfract7070495 doi: 10.3390/fractalfract7070495
[12]	J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, M. A. Taneco-Hernandez, G. V. Guerrero-Ramírez, Electrical circuits RC and RL involving fractional operators with bi-order, Adv. Mech. Eng., 9 (2017), 6. https://doi.org/10.1177/1687814017707132 doi: 10.1177/1687814017707132
[13]	J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, C. M. Astorga-Zaragoza, J. Reyes-Reyes, Analytical and numerical solutions of electrical circuits described by fractional derivatives, Appl. Math. Model., 40 (2016), 9079–9094. https://doi.org/10.1016/j.apm.2016.05.041 doi: 10.1016/j.apm.2016.05.041
[14]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000.
[15]	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 (2009), 299–318.
[16]	S. Hristova, A. Benkerrouche, M. Said Souid, A. Hakem, Boundary value problems of Hadamard fractional differential equations of variable order, Symmetry, 13 (2021), 896. https://doi.org/10.3390/sym13050896 doi: 10.3390/sym13050896
[17]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, elsevier, 2006.
[18]	G. Li, Y. Zhang, Y. Guan, W. Li, Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse, Math. Biosci. Eng., 20 (2023), 7020–7041. https://doi.org/10.3934/mbe.2023303. doi: 10.3934/mbe.2023303
[19]	M. Malarvizhi, S. Karunanithi, N. Gajalakshmi, Numerical analysis using RK-4 in transient analysis of rlc circuit, Adv. Math. Sci. J., 9 (2020), 6115–6124, 2020. https://doi.org/10.37418/amsj.9.8.79 doi: 10.37418/amsj.9.8.79
[20]	S. Muthaiah, M. Murugesan, N. G. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theor. Nonlinear Anal. Appl., 3 (1019), 162–173. https://doi.org/10.31197/atnaa.579701 doi: 10.31197/atnaa.579701
[21]	C. Promsakon, S. K. Ntouyas, J. Tariboon, Hilfer-Hadamard nonlocal integro-multipoint fractional boundary value problems, J. Funct. Space., 2021 (2021), 8031524. https://doi.org/10.1155/2021/8031524 doi: 10.1155/2021/8031524
[22]	A. Y. A. Salamooni, D. D. Pawar, Existence and uniqueness of nonlocal boundary conditions for Hilfer-Hadamard-type fractional differential equations, Adv. Differ. Equ., 2021 (2021), 198. https://doi.org/10.1186/s13662-021-03358-0 doi: 10.1186/s13662-021-03358-0
[23]	A. A. kilbas, O. I. Marichev, S. G. Samko, Fractional intearals and derivatives: theorv and agglications, 1993.
[24]	Y. Sang, L. He, Y. Wang, Y. Ren, N. Shi, Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed-point theorem, Adv. Differ. Equ., 2021 (2021), 156. https://doi.org/10.1186/s13662-021-03318-8 doi: 10.1186/s13662-021-03318-8
[25]	C. Wang, T. Z. Xu, Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60 (2015), 383–393. https://doi.org/10.1007/s10492-015-0102-x doi: 10.1007/s10492-015-0102-x
[26]	A. Zada, W. Ali, S. Farina, Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Method. Appl. Sci., 40 (2017), 5502–5514. https://doi.org/10.1002/mma.4405 doi: 10.1002/mma.4405

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