Wavelet methods for fractional electrical circuit equations

Abstract

Classical electric circuits consists of resistors, inductors and capacitors which have irreversible and lossy properties that are not taken into account in classical analysis. FDEs can be interpreted as basic memory operators and are generally used to model the lossy properties or defects. Therefore, employing fractional differential terms in electric circuit equations provides accurate modelling of those circuit elements. In this paper, the numerical solutions of fractional LC, RC and RLC circuit equations are considered to better model those imperfections. To this end, the operational matrices for Bernoulli and Chebyshev wavelets are used to obtain the numerical solutions of those fractional circuit equations. Chebyshev wavelets are orthogonal, and under some circumstances, Bernoulli wavelets can be orthogonal. The wavelet methods' quick convergence and minimal processing load depend on the orthogonality principle. In the proposed method, those FDEs are transformed into algebraic equation systems using operational matrices employing the discrete Wavelets. The performance of those two wavelet methods are compared and contrasted for computational load, speed, and absolute error values. The paper exploits discrete Bernoulli and Chebyshev wavelets for the numerical solution of fractional LC, RC and RLC circuit equations. The fast convergence, low processing burden, and compactness of the Bernoulli and Chebyshev wavelet methods for fractional circuit equation solutions represent the novel contributions of this paper. Numerical solutions and comparisons are also presented to validate the method.

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

“…The analysis is conducted under a fractional derivative order of µ = 0.5. The resulting outcomes are visually compared with established techniques, such as the Chebyshev Wavelets of the third kind Method (Ch3WM) [24], in Figure 5. Given that the exact solution of the RC circuit is defined for integer derivative orders, rendering it unsuitable as a reference under fractional order µ = 0.5, we employ the Adams-Bashforth method (ABM) [35] as a reference technique.…”

“…In Figure 8, we modify the configuration for the fractional-order LC circuit by setting µ = 1.5 and applying GLOMM. The resulting outcomes are visually contrasted with well-established techniques, such as the Bernoulli Wavelet Method (BWM) [24]. Considering that the exact solution of the LC circuit is defined for integer derivative orders, making it unsuitable as a reference under fractional order µ = 1.5, we again resort to the ABM [35] as a reference technique.…”

“…Confronted with the inherent difficulty of obtaining exact analytic solutions for nonlinear FDEs, researchers have developed an arsenal of numerical and approximate methods. In addition to spectral collocation [19], variational iteration [20], differential quadrature [21], adomian decomposition [22], fractional reduced differential transform [23], and wavelet methods [24][25][26], innovative techniques such as finite element method [27], and radial basis function methods [28] have been meticulously crafted to surmount these challenges. On the other hand, many numerical techniques for solving fractional models face limitations, including accuracy issues with complex dynamics and non-standard boundaries, computational inefficiency for large-scale simulations, and restricted applicability to specific equations or systems.…”