Semi-Analytical Solutions of Fractional Differential 
Equations in RL and RC Circuits

V. D. Mathpati; B. B. Pandit

doi:https://doi.org/10.14445/22315373/IJMTT-V72I3P103

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 72 | Issue 3 | Year 2026 | Article Id. IJMTT-V72I3P103 | DOI : https://doi.org/10.14445/22315373/IJMTT-V72I3P103

Semi-Analytical Solutions of Fractional Differential Equations in RL and RC Circuits

V. D. Mathpati, B. B. Pandit

Received	Revised	Accepted	Published
16 Jan 2026	21 Feb 2026	12 Mar 2026	26 Mar 2026

Citation :

V. D. Mathpati, B. B. Pandit, "Semi-Analytical Solutions of Fractional Differential Equations in RL and RC Circuits," International Journal of Mathematics Trends and Technology (IJMTT), vol. 72, no. 3, pp. 13-22, 2026. Crossref, https://doi.org/10.14445/22315373/IJMTT-V72I3P103

Abstract

Fractional-order differential equations provide an effective mathematical model for electrical circuits that exhibit memory and nonlocal effects. This study investigates semi-analytical methods to solve fractional-order differential equations of RL and RC electrical circuits. The semi-analytical methods, namely the Generalized Differential Transform Method (GDTM), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), and Adomian Decomposition Method (ADM), are used to solve equations. The obtained series solutions are analyzed and compared graphically for different values of the fractional-order to illustrate their impact on circuit dynamics. The comparative results provide useful guidance for selecting an appropriate mathematical method for solving fractional-order electrical circuit models.

Keywords

Fractional-Order Circuits, Caputo Fractional Derivative, Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), Generalized Differential Transform Method (GDTM).

References

[1] I. Podlubny, Fractional Differential Equations, New York, USA: Academic Press, 1999.
[Google Scholar] [Publisher Link]

[2] Keith B. Oldham, and Jerome Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, New York, USA: Academic Press, 1974.
[Google Scholar] [Publisher Link]

[3] Ivo Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Berlin, Germany: Springer, 2010.
[Google Scholar] [Publisher Link]

[4] Tadeusz Kaczorek, and Krzysztof Rogowski, Fractional Linear Systems and Electrical Circuits, Berlin, Germany: Springer, 2015.
[CrossRef] [Google Scholar] [Publisher Link]

[5] A.G. Radwan, and K.N. Salama, “Fractional-Order RC and RL Circuits,” Circuits, Systems and Signal Processing, vol. 31, pp. 1901–1915, 2012.
[CrossRef] [Google Scholar] [Publisher Link]

[6] Gómez-Aguilar Jose Francisco et al., “Fractional RC and LC Electrical Circuits Circuitos Electricos RC y LC Fraccionarios,” Ingeniería Investigación y Tecnología, vol. 15, no. 2, pp. 311–319, 2014.
[CrossRef] [Google Scholar] [Publisher Link]

[7] J.F. Gómez-Aguilar et al., “Electrical Circuits Described by a Fractional Derivative with Regular Kernel,” Revista Mexicana de Física, vol. 62, no. 2, 2016.
[Google Scholar] [Publisher Link]

[8] Jose Francisco Gómez-Aguilar et al., “Analytical Solutions of the Electrical RLC Circuit via Liouville–Caputo Operators,” Entropy, vol. 18, no. 8, 2016.
[CrossRef] [Google Scholar] [Publisher Link]

[9] J.F. Gómez-Aguilar, Abdon Atangana, and V.F. Morales-Delgado, “Electrical Circuits RC, LC and RL Described by Atangana–Baleanu Fractional Derivatives,” International Journal of Circuit Theory and Applications, vol. 45, no. 11, pp. 1514–1533, 2017.
[CrossRef] [Google Scholar] [Publisher Link]

[10] V.F. Morales-Delgado, J.F. Gómez-Aguilar, and M.A. Taneco-Hernández, “Analytical Solutions of Electrical Circuits Described by Fractional Conformable Derivatives in Liouville-Caputo Sense,” AEU – International Journal of Electronics and Communications, vol. 85, pp. 108–117, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[11] Ndolane Sene, and J.F. Gómez-Aguilar, “Analytical Solutions of Electrical Circuits Considering Certain Generalized Fractional Derivatives,” European Physical Journal Plus, vol. 134, 2019.
[CrossRef] [Google Scholar] [Publisher Link]

[12] J.F. Gómez-Aguilar, “Fundamental Solutions to Electrical Circuits of Non-integer Order Via Fractional Derivatives with and without Singular Kernels,” European Physical Journal Plus, vol. 133, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[13] Kashif Ali Abro, Anwar Ahmed Atangana, and Muhammad Aslam Uqaili, “Comparative Mathematical Analysis of RL and RC Electrical Circuits Via Atangana–Baleanu and Caputo–Fabrizio Fractional derivatives,” The European Physical Journal Plus, vol. 133, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[14] P.V. Shah et al., “Analytic Solution for the RL Electric Circuit Model in Fractional Order,” Abstract and Applied Analysis, 2014.
[CrossRef] [Google Scholar] [Publisher Link]

[15] N. Magesh, and A. Saravanan, “Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-integer Order,” Nonlinear Engineering, vol. 7, no. 2, pp. 127-135, 2018.
[CrossRef] [Google Scholar] [Publisher Link]

[16] Marwan Abukhaled, and S.A. Khuri, “RLC Electric Circuit Model of Fractional Order: A Green’s Function Approach,” International Journal of Computer Mathematics, vol. 101, no. 9-10, pp. 961-972, 2023.
[CrossRef] [Google Scholar] [Publisher Link]

[17] Sowmya Muniswamy, and Vidya Patil, “Numerical Analysis of RLC Electrical Circuits using Caputo Fractional Differential Operator,” ECS Transactions, vol. 107, no. 1, 2022.
[CrossRef] [Google Scholar] [Publisher Link]

[18] Changdev P. Jadhav et al., “Some problems of Mesh R–L electrical Circuits using Fractional Calculus,” Alexandria Engineering Journal, vol. 115, pp. 111-118, 2025.
[CrossRef] [Google Scholar] [Publisher Link]

[19] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
[CrossRef] [Google Scholar] [Publisher Link]

[20] Yizheng Hu, Yong Luo, and Zhengyi Lu, “Analytical Solution of Linear Fractional Differential Equation by Adomian Decomposition Method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 220–229, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[21] Jin-Fa Cheng, and Yu-Ming Chu, “Solution to the Linear Fractional Differential Equation using Adomian Decomposition Method,” Mathematical Problems in Engineering, 2011.
[CrossRef] [Google Scholar] [Publisher Link]

[22] Ji-Huan He, “Homotopy Perturbation Technique,” Computer Method in Applied Mechanics and Engineering, vol. 178, no. 3–4, pp. 257–262, 1999.
[CrossRef] [Google Scholar] [Publisher Link]

[23] O. Abdulaziz, I. Hashim, and S. Momani, “Application of Homotopy-Perturbation Method to Fractional IVPs,” Journal of Computational and Applied Mathematics, vol. 216, no. 2, pp. 574–584, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[24] Shumaila Javeed et al., “Analysis of homotopy Perturbation Method for Solving Fractional Order Differential Equations,” Mathematics, vol. 7, no. 1, 2019.
[CrossRef] [Publisher Link]

[25] Zaid Odibat, and Shaher Momani, “A Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194-199, 2008.
[CrossRef] [Google Scholar] [Publisher Link]

[26] Zaid M. Odibat et al., “A Study on the Convergence Conditions of Generalized Differential Transform Method,” Mathematical Methods in the Applied Sciences, vol. 40, no. 1, pp. 40-48, 2017.
[CrossRef] [Google Scholar] [Publisher Link]

[27] Ji-Huan He, and Xu-Hong Wu, “Variational Iteration Method: New Development and Applications,” Computers and Mathematics with Applications, vol. 54, no. 7–8, pp. 881–894, 2007.
[CrossRef] [Google Scholar] [Publisher Link]

[28] Ji-Huan He, “A Short Remark on Fractional Variational Iteration Method,” Physics Letters A, vol. 375, no. 38, pp. 3362–3364, 2011.
[CrossRef] [Google Scholar] [Publisher Link]

[29] Shuiping Yang, Aiguo Xiao, and Hong Su, “Convergence of the Variational Iteration Method for Solving Multi-order Fractional Differential Equations,” Computers and Mathematics with Applications, vol. 60, no. 10, pp. 2871–2879, 2010.
[CrossRef] [Google Scholar] [Publisher Link]

[30] M. Fourier, Théorie Analytique de la Chaleur, Paris, France: Firmin Didot, 1822.
[Google Scholar] [Publisher Link]