Exploring dynamics in RLC circuits: a novel approach utilizing the $ (k, \phi) $-Hilfer proportional fractional operator

Weerawat Sudsutad; Aphirak Aphithana; Chatthai Thaiprayoon; Jutarat Kongson; Weerawat Sudsutad; Aphirak Aphithana; Chatthai Thaiprayoon; Jutarat Kongson

doi:10.3934/math.20251003

AIMS Mathematics

2025, Volume 10, Issue 9: 22531-22560. doi: 10.3934/math.20251003

Previous Article Next Article

Research article Special Issues

Exploring dynamics in RLC circuits: a novel approach utilizing the $ (k, \phi) $-Hilfer proportional fractional operator

1.
Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand
2.
Division of Mathematics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Krungthep, Bangkok 10120, Thailand
3.
Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

Received: 01 August 2025 Revised: 23 September 2025 Accepted: 25 September 2025 Published: 29 September 2025
MSC : 00A71, 34A05, 34A08, 34B15, 47H10

In this study, the theory of fractional calculus is applied to the electrical circuits. In this work, we investigated the Langevin-type differential equations under the $ (k, \phi) $-Hilfer proportional fractional derivative. By utilizing the bivariate Mittag-Leffler function and the $ \psi $-Laplace transform, we designed a representation of an explicit analytical solution for the linear system corresponding to the considered model. We explored Ulam–Hyers stability results with the Mittag-Leffler function and their generalizations to confirm Ulam stability by applying the extended Gronwall inequality under the context of the $ (k, \phi) $-proportional fractional operators. Finally, the RLC electrical circuit model was chosen as the application's agent to validate the accuracy of our theoretical results. Our results offer additional analytical choices due to the wider range of parameter values compared to earlier studies.
- electrical circuit,
- fractional differential equation,
- $ \phi $-Laplace transform,
- fractional RLC circuit,
- $ (k, \phi) $-Hilfer proportional derivative
Citation: Weerawat Sudsutad, Aphirak Aphithana, Chatthai Thaiprayoon, Jutarat Kongson. Exploring dynamics in RLC circuits: a novel approach utilizing the $ (k, \phi) $-Hilfer proportional fractional operator[J]. AIMS Mathematics, 2025, 10(9): 22531-22560. doi: 10.3934/math.20251003

Related Papers:

Abstract

In this study, the theory of fractional calculus is applied to the electrical circuits. In this work, we investigated the Langevin-type differential equations under the $ (k, \phi) $-Hilfer proportional fractional derivative. By utilizing the bivariate Mittag-Leffler function and the $ \psi $-Laplace transform, we designed a representation of an explicit analytical solution for the linear system corresponding to the considered model. We explored Ulam–Hyers stability results with the Mittag-Leffler function and their generalizations to confirm Ulam stability by applying the extended Gronwall inequality under the context of the $ (k, \phi) $-proportional fractional operators. Finally, the RLC electrical circuit model was chosen as the application's agent to validate the accuracy of our theoretical results. Our results offer additional analytical choices due to the wider range of parameter values compared to earlier studies.

References

[1]	I. Podlubny, Fractional differential equations, New York, NY, USA, 1998.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, North-Holland Mathematics Studies, Amsterdam, The Netherlands: Elsevier, 2006.
[3]	J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus, Springer, 2007. https://doi.org/10.1007/978-1-4020-6042-7
[4]	J. V. Devi, S. Leela, Theory of fractional dynamic systems, Cambridge, UK: Cambridge Scientific Publishers, 2009.
[5]	K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, New York, NY, USA: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[6]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014.
[7]	R. Hilfer, Applications of fractional calculus in physics, Sigapore: World Scientific, 2000. https://doi.org/10.1142/3779
[8]	S. Mebeen, G. M. Habibullah, $k$–fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[9]	G. A. Dorrego, An alternative definition for the $k$–Riemann–Liouville fractional derivative, Appl. Math. Sci., 9 (2015), 481–491. https://doi.org/10.12988/ams.2015.411893 doi: 10.12988/ams.2015.411893
[10]	Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, S. M. Kang, Generalized Riemann–Liouville $k$–fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access, 6 (2018), 64946–64953. https://doi.org/10.1109/ACCESS.2018.2878266 doi: 10.1109/ACCESS.2018.2878266
[11]	J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the $\psi$–Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[12]	F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function, Adv. Differ. Equ., 2020 (2020), 303. https://doi.org/10.1186/s13662-020-02767-x doi: 10.1186/s13662-020-02767-x
[13]	I. Ahmad, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Differ. Equ., 2020 (2020), 329. https://doi.org/10.1186/s13662-020-02792-w doi: 10.1186/s13662-020-02792-w
[14]	K. D. Kucche, A. D. Mali, On the nonlinear $(k, \psi)$–Hilfer fractional differential equations, Chaos Soliton. Fract., 152 (2021), 111335. https://doi.org/10.1016/j.chaos.2021.111335 doi: 10.1016/j.chaos.2021.111335
[15]	W. Sudsutad, J. Kongson, C. Thaiprayoon, On generalized $(k, \psi)$-Hilfer proportional fractional operator and its applications to the higher-order Cauchy problem, Bound. Value Probl., 2024 (2024), 83. https://doi.org/10.1186/s13661-024-01891-x doi: 10.1186/s13661-024-01891-x
[16]	W. Sudsutad, C. Thaiprayoon, A. Aphithana, J. Kongson, W. Sae-dan, Qualitative results and numerical approximations of the $(k, \psi)$–Caputo proportional fractional differential equations and applications to blood alcohol levels model, AIMS Math., 9 (2024), 34013–34041. https://doi.org/10.3934/math.20241622 doi: 10.3934/math.20241622
[17]	H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional calculus and fractional processes with applications to financial economics, London, UK: Academic Press, 2016.
[18]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, Singapore: World Scientific, 2012.
[19]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity, London: Imperial College Press, 2010.
[20]	D. S. Lemons, A. Gythiel, Paul Langevin's 1908 paper "On the Theory of Brownian Motion" ["Sur la théorie du mouvement brownien, " C. R. Acad. Sci. (Paris) 146,530–533 (1908)], Am. J. Phys., 65 (1997), 1079–1081. https://doi.org/10.1119/1.18725 doi: 10.1119/1.18725
[21]	B. Li, S. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput., 53 (2017), 683–692. https://doi.org/10.1007/s12190-016-0988-9 doi: 10.1007/s12190-016-0988-9
[22]	H. Baghani, J. J. Nieto, On fractional Langevin equation involving two fractional orders in different intervals, Nonlinear Anal.: Model. Control, 24 (2019), 884–897. https://doi.org/10.15388/NA.2019.6.3 doi: 10.15388/NA.2019.6.3
[23]	M. Aydin, N. I. Mahmudov, The sequential conformable Langevin-type differential equations and their applications to the RLC electric circuit problems, J. Appl. Math., 2024 (2024), 3680383. https://doi.org/10.1155/2024/3680383 doi: 10.1155/2024/3680383
[24]	T. Yu, K. Deng, M. Luo, Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1661–1668. https://doi.org/10.1016/j.cnsns.2013.09.035 doi: 10.1016/j.cnsns.2013.09.035
[25]	H. Zhou, J. Alzabut, L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, Eur. Phys. J. Spec. Top., 226 (2017), 3577–3590. https://doi.org/10.1140/epjst/e2018-00082-0 doi: 10.1140/epjst/e2018-00082-0
[26]	O. Baghani, On fractional Langevin equation involving two fractional orders, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 675–681. https://doi.org/10.1016/j.cnsns.2016.05.023 doi: 10.1016/j.cnsns.2016.05.023
[27]	F. Norouzi, G. M. N'Guérékata, A study of $\psi$-Hilfer fractional differential system with application in financial crisis, Chaos, Soliton. Fract.: X, 6 (2021), 100056. https://doi.org/10.1016/j.csfx.2021.100056 doi: 10.1016/j.csfx.2021.100056
[28]	A. Prakash, N. Kalyan, S. Ahuja, Analysis and numerical simulation of fractional-order blood alcohol model with singular and non-singular kernels, Comput. Math. Biophys., 12 (2024), 20240001. https://doi.org/10.1515/cmb-2024-0001 doi: 10.1515/cmb-2024-0001
[29]	N. Hatime, S. Melliani, A. El Mfadel, M. H. Elomari, Numerical analysis of generalized fractional form of Newton's cooling law under a variable environment temperature, Int. J. Appl. Comput. Math., 10 (2024), 61. https://doi.org/10.1007/s40819-024-01705-9 doi: 10.1007/s40819-024-01705-9
[30]	E. Fendzi-Donfack, J. P. Nguenang, L. Nana, Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation, Eur. Phys. J. Plus, 133 (2018), 32. https://doi.org/10.1140/epjp/i2018-11851-1 doi: 10.1140/epjp/i2018-11851-1
[31]	E. Fendzi-Donfack, J. P. Nguenang, L. Nana, On the soliton solutions for an intrinsic fractional discrete nonlinear electrical transmission line, Nonlinear Dyn., 104 (2021), 691–704. https://doi.org/10.1007/s11071-021-06300-x doi: 10.1007/s11071-021-06300-x
[32]	S. M. Ulam, A collection of mathematical problems, New York: Interscience, 1960.
[33]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[34]	K. D. Kucche, J. P. Kharade, Global existence and Ulam-Hyers stability of $\psi$-Hilfer fractional differential equations, Kyungpook Math. J., 60 (2020), 647–671. https://doi.org/10.5666/KMJ.2020.60.3.647 doi: 10.5666/KMJ.2020.60.3.647
[35]	T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297–300. https://doi.org/10.2307/2042795 doi: 10.2307/2042795
[36]	J. Vanterler da C Sousa, E. C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi$-Hilfer operator, J. Fixed Point Theory Appl., 20 (2018), 96.
[37]	K. Liu, J. Wang, D. O'Regan, Ulam-Hyers-Mittag-Leffler stability for $\psi$-Hilfer fractional-order delay differential equations, Adv. Differ. Equ., 2019 (2019), 50. https://doi.org/10.1186/s13662-019-1997-4 doi: 10.1186/s13662-019-1997-4
[38]	W. Sudsutad, C. Thaiprayoon, B. Khaminsou, J. Alzabut, J. Kongson, A Gronwall inequality and its applications to the Cauchy-type problem under $\psi$-Hilfer proportional fractional operators, J. Inequal. Appl., 2023 (2023), 20. https://doi.org/10.1186/s13660-023-02929-x doi: 10.1186/s13660-023-02929-x
[39]	M. Awadalla, Y. Y. Y. Noupoue, K. A. Asbeh, Psi-Caputo logistic population growth model, J. Math., 2021 (2021), 8634280. https://doi.org/10.1155/2021/8634280 doi: 10.1155/2021/8634280
[40]	M. Samraiz, Z. Perveen, G. Rahman, K. S. Nisar, D. Kumar, On the $(k, s)$-Hilfer-Prabhakar fractional derivative with applications to mathematical physics, Front. Phys., 8 (2020), 309. https://doi.org/10.3389/fphy.2020.00309 doi: 10.3389/fphy.2020.00309
[41]	B. Mohammadaliee, V. Room, M. E. Samei, SEIARS model for analyzing COVID-19 pandemic process via $\psi$-Caputo fractional derivative and numerical simulation, Sci. Rep., 14 (2024), 723. https://doi.org/10.1038/s41598-024-51415-x doi: 10.1038/s41598-024-51415-x
[42]	F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst., Ser. S, 13 (2020), 709–722. https://doi.org/10.3934/dcdss.2020039 doi: 10.3934/dcdss.2020039
[43]	M. L. Abell, J. P. Braselton, Introductory differential equations with boundary value problems, Academic Press, 2010.
[44]	A. Ahmadova, N. I. Mahmudov, Langevin differential equations with general fractional orders and their applications to electric circuit theory, J. Comput. Appl. Math., 388 (2021), 113299. https://doi.org/10.1016/j.cam.2020.113299 doi: 10.1016/j.cam.2020.113299
[45]	S. Naveen, V. Parthiban, M. I. Abbas, Qualitative analysis of RLC circuit described by Hilfer derivative with numerical treatment using the Lagrange polynomial method, Fractal Fract., 7 (2023), 804, https://doi.org/10.3390/fractalfract7110804 doi: 10.3390/fractalfract7110804
[46]	M. Murugesan, S. Shanmugam, M. Rhaima, R. Ravi, Analysis of non-local integro-differential equations with Hadamard fractional derivatives: existence, uniqueness, and stability in the context of RLC models, Fractal Fract., 8 (2024), 409. https://doi.org/10.3390/fractalfract8070409 doi: 10.3390/fractalfract8070409
[47]	R. G. Thunibat, A. N. Abdulrahman, E. K. Jaradat, O. K. Jaradat, Demonstrating nonlinear RLC circuit equation by involving fractional adomian decomposition method by Caputo definition, Adv. Math. Models Appl., 9 (2024), 387–400. https://doi.org/10.62476/amma93387 doi: 10.62476/amma93387
[48]	M. Murugesan, R. Meganathan, R. S. Shanth, M. Rhaima, Existence and analysis of Hilfer-Hadamard fractional differential equations in RLC circuit models, AIMS Math., 9 (2024), 28741–28764. https://doi.org/10.3934/math.20241394 doi: 10.3934/math.20241394
[49]	M. Shankar, S. N. Bora, Generalized Ulam-Hyers-Rassias stability of solution for the Caputo fractional non-instantaneous impulsive integro-differential equation and its application to fractional RLC circuit, Circuits, Syst. Signal Process., 42 (2023), 1959–1983. https://doi.org/10.1007/s00034-022-02217-x doi: 10.1007/s00034-022-02217-x

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)