Spectral collocation with generalized Laguerre operational matrix for numerical solutions of fractional electrical circuit models

Year 2024, , 110 - 132, 31.03.2024

İbrahim Avcı

https://doi.org/10.53391/mmnsa.1428035

Cited By: 1

Abstract

In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor ($RL$), Resistor-Capacitor ($RC$), Resonant (Inductor-Capacitor) ($LC$), and Resistor-Inductor-Capacitor ($RLC$) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models.

Keywords

numerical analysis, electrical circuits, generalized Laguerre polynomials, fractional integrals, fractional derivatives

References

• [1] Machado, J.T., Kiryakova, V. and Mainardi, F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 16(3), 1140-1153, (2011).
• [2] Miller, K.S. and Rosso, B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley: New York, (1993).
• [3] Oldham, K.B; Spanier, J. The Fractional Calculus; Academic Press: New York, USA, (1974).
• [4] Caputo, M. Linear models of dissipation whose Q is almost frequency independent—II. Geophysical Journal International, 13(5), 529-539, (1967).
• [5] Debnath, L. Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003, 3413-3442, (2003).
• [6] Tarasov, V.E. Mathematical economics: application of fractional calculus. Mathematics, 8(5), 660, (2020).
• [7] Alinei-Poiana, T., Dulf, E.H. and Kovacs, L. Fractional calculus in mathematical oncology. Scientific Reports, 13, 10083, (2023).
• [8] Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific: Singapore, (2022).
• [9] Avcı, I., Lort, H. and Tatlıcıoglu, B.E. Numerical investigation and deep learning approach for fractal–fractional order dynamics of Hopfield neural network model. Chaos, Solitons & Fractals, 177, 114302, (2023).
• [10] Chen, S.B., Soradi-Zeid, S., Jahanshahi, H., Alcaraz, R., Gómez-Aguilar, J.F., Bekiros, S. et al. Optimal control of time-delay fractional equations via a joint application of radial basis functions and collocation method. Entropy, 22(11), 1213, (2020).
• [11] Joshi, H. and Yavuz, M. Transition dynamics between a novel coinfection model of fractional-order for COVID-19 and tuberculosis via a treatment mechanism. The European Physical Journal Plus, 138, 468, (2023).
• [12] Soradi-Zeid, S., Jahanshahi, H., Yousefpour, A. and Bekiros, S. King algorithm: a novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems. Chaos, Solitons & Fractals, 132, 109569, (2020).
• [13] Rezapour, S., Asamoah, J.K.K., Etemad, S., Akgül, A., Avcı, I. and El Din, S.M. On the fractal-fractional Mittag-Leffler model of a COVID-19 and Zika Co-infection. Results in Physics, 55, 107118, (2023).
• [14] Zhao, W., Leng, K., Chen, J., Jiao, Y. and Zhao, Q. Research on statistical algorithm optimization of fractional differential equations of quantum mechanics in ecological compensation. The European Physical Journal Plus, 134, 316, (2019).
• [15] Duran, S., Durur, H., Yavuz, M. and Yokus, A. Discussion of numerical and analytical techniques for the emerging fractional order murnaghan model in materials science. Optical and Quantum Electronics, 55, 571, (2023).
• [16] Yilmaz, B. A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus. Optik, 247, 168026, (2021).
• [17] Barros, L.C.D., Lopes, M.M., Pedro, F.S., Esmi, E., Santos, J.P.C.D. Sánchez, D.E. et al. The memory effect on fractional calculus: an application in the spread of COVID-19. Computational and Applied Mathematics, 40, 72, (2021).
• [18] Tarasov, V.E. On history of mathematical economics: application of fractional calculus. Mathematics, 7(6), 509, (2019).
• [19] El-Gamel, M., Mohamed, N. and Waleed, A. Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis. Mathematical Modelling and Numerical Simulation with Applications, 3(4), 351-375, (2023).
• [20] Wu, G.C. A fractional variational iteration method for solving fractional nonlinear differential equations. Computers & Mathematics with Applications, 61(8), 2186-2190, (2011).
• [21] Mohamed, S.A. A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems. Mathematical Methods in the Applied Sciences, (2020).
• [22] Albogami, D., Maturi, D. and Alshehri, H. Adomian decomposition method for solving fractional Time-Klein-Gordon equations using Maple. Applied Mathematics, 14(6), 411-418, (2023).
• [23] Abuasad, S., Hashim, I. and Abdul Karim, S.A. Modified fractional reduced differential transform method for the solution of multiterm time-fractional diffusion equations. Advances in Mathematical Physics, 2019, 5703916, (2019).
• [24] Tural-Polat, S.N. and Dincel, A.T. Wavelet methods for fractional electrical circuit equations. Physica Scripta, 98(11), 115203, (2023).
• [25] Yadav, P., Jahan, S. and Nisar, K.S. Shifted fractional order Gegenbauer wavelets method for solving electrical circuits model of fractional order. Ain Shams Engineering Journal, 14(11), 102544, (2023).
• [26] Ahmed, S., Shah, K., Jahan, S. and Abdeljawad, T. An efficient method for the fractional electric circuits based on Fibonacci wavelet. Results in Physics, 52, 106753, (2023).
• [27] Li, M., Huang, C. and Wang, P. Galerkin finite element method for nonlinear fractional Schrödinger equations. Numerical Algorithms, 74, 499-525, (2017).
• [28] Zafarghandi, F.S., Mohammadi, M., Babolian, E. and Javadi, S. Radial basis functions method for solving the fractional diffusion equations. Applied Mathematics and Computation, 342, 224-246, (2019).
• [29] Alexander, C.K. Fundamentals of Electric Circuits. McGraw-Hill, (2013).
• [30] Kaczorek, T. and Rogowski, K. Positive fractional electrical circuits. In Fractional Linear Systems and Electrical Circuits (Vol. 13) (pp. 49-80). Switzerland: Springer Cham, (2015).
• [31] Ibrahim Nuruddeen, R., Gómez-Aguilar, J.F., Garba Ahmad, A. and Ali, K.K. Investigating the dynamics of Hilfer fractional operator associated with certain electric circuit models. International Journal of Circuit Theory and Applications, 50(7), 2320-2341, (2022).
• [32] Bhrawy, A.H., Baleanu, D., Assas, L.M. and Tenreiro Machado, J.A. On a generalized Laguerre operational matrix of fractional integration. Mathematical Problems in Engineering, 2013, 569286, (2013).
• [33] Dimitrov, D.K., Marcellán, F. and Rafaeli, F.R. Monotonicity of zeros of Laguerre–Sobolev-type orthogonal polynomials. Journal of Mathematical Analysis and Applications, 368(1), 80-89, (2010).
• [34] Avcı, I. Numerical simulation of fractional delay differential equations using the operational ˙ matrix of fractional integration for fractional-order Taylor basis. Fractal and Fractional, 6(1), 10, (2021).
• [35] Diethelm, K., Ford, N.J. and Freed, A.D. Detailed error analysis for a fractional Adams method. Numerical Algorithms, 36, 31-52, (2004).