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

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.