Wavelet methods for fractional electrical circuit equations

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.


Introduction
Researchers have become more interested in the use of fractional calculus and Fractional Differential Equations in recent years.FDE is the extension of common integer order integrals and derivatives to arbitrary orders.Despite the fact that this subject is almost as old as conventional calculus, the notions seem rather complex and difficult to understand.On the other hand, for the inherited characteristics of various materials and processes, FDE systems and models are useful because those can take into account the effects that cannot be modeled with traditional integer-order systems.Thus, FDEs are used in a variety of engineering and applied science disciplines such as chaos theory, signal processing, optimal control, quantum mechanics, electromagnetic theory and so on [1][2][3][4][5][6][7][8][9][10][11][12][13][14].However, the same complex modelling feature renders finding the analytical solutions to those FDEs very difficult, and therefore it is crucial to have accurate numerical solutions.In recent years, certain numerical methods have been used for FDEs, such as spectral method [15], Galerkin method [16], Homotopy analysis method [17], Jacobi Tau method [18], waveform relaxation method [19], fractional finite difference method [20], Adomian decomposition method [21,22], Fourier transform [23], power series method [24], B-spline collocation method [25], block-by-block method [26], variational iteration method [27], polynomial methods [28][29][30][31][32] and wavelet methods [33][34][35][36][37][38][39].Wavelets are mathematical functions that can be used to decompose data into various time-frequency components.These functions are obtained by shifting and scaling of a so called 'mother' wavelet function.The essential contributions of the wavelet basis can be summarized as the transformation of the FDE into an algebraic equation, which in turn can be solved numerically, and also the fast and easy convergence of the numerical method thanks to the orthogonality, compact support, singularity detection, and the simultaneous multiple resolution properties of the wavelets.
Classical electrical circuits consist of resistors, inductors and capacitors.These elements also contain lossy and irreversible features that are not taken into account in the classical analysis.Fractional differential terms can also be interpreted as memory operators and are often used to model lossy effects or damage.The generalization of electrical signal propagation and more accurate modeling of circuit components are made possible by the use of fractional differential terms in electrical circuits [40][41][42].The Laplace transform method [43,44], Sumudu transform method [45], Legendre Wavelet Method using Riemann-Liouville fractional derivative definition [46], and Haar wavelet method using Caputo fractional derivative definition [47] are the numerical solution methods for fractional circuit equations developed up to now.The equations created by using the Caputo fractional derivative expression are subjected to a numerical Laplace transform by Gomez et al [43], and their solutions in the time domain are derived in terms of the Mittag-Leffler function.For fractional derivatives calculated using the Caputo definition in RL electric circuits, Shah et al [44] employ the Laplace transform.In another study, Gill et al [45] obtain the solutions of fractional RLC circuits in terms of Mittag-Leffler function using Sumudu transform.Arora and Chauan [46] use the Legendre wavelet method for the solution of these fractional circuit equations, while Altaf and Khan [47] examine the Haar wavelet method for similar fractional RLC circuit solutions.
In this study, approximate solutions of fractional order LC, RC and RLC circuit equations given in equations (1)-( 3) [47] with two wavelet methods are discussed.
To this end, numerical solutions of the fractional order LC, RC and RLC circuit equations are obtained by using the operational matrix approach of Bernoulli and Chebyshev wavelets of the third kind mainly because of their orthogonality and compact support properties.The wavelet methods for numerical solutions of FDEs generally use operational matrices for fractional integration to convert the original FDE into a discrete-time algebraic equation, which in turn can easily be solved using numerical methods.Orthogonality principle is crucial for sparse operational matrices, which provides quick convergence and requires minimal computational load.Operational Matrices for the numerical integration is formulized for Bernoulli Wavelets (Bernoulli Wavelet Method, BWM) and Chebyshev Wavelets of the third kind (Chebyshev Wavelets of the third kind Method, Ch3WM) and used to solve fractional LC, RC and RLC circuit equations for the first time in this paper.Comparisons of the obtained numerical solutions are presented.The paper is organized as in the following.In section 2, the definitions for fractional calculus are given.In section 3, The Bernoulli and Chebyshev wavelets are defined.The operational matrices for Bernoulli and Chebyshev wavelets are obtained in section 4. The proposed method is presented in section 5. Numerical solutions for several FDEs are provided in section 6.The paper is concluded in section 7.

Foundations
The preliminary definitions for fractional calculus that are utilized in the paper are presented in this section.

Definition 1
Riemann-Liouville and Caputo definitions for fractional derivatives are most commonly used ones among many others [48].The Riemann-Liouville fractional integral operator of order α 0 is defined by [48,49] Riemann-Liouville derivatives possess some disadvantages to model real world phenomena with FDEs.Therefore a modified fractional differential operator a D f is more commonly used which is proposed by Caputo.

Definition 2
The Caputo definition of fractional derivative operator is given by [48,49] ò The following expressions relate the Riemann-Liouville operator and Caputo operator: = -

Bernoulli and chebyshev wavelets
A collection of wavelike functions are referred to as wavelets.By dilating and translating a so-called mother wavelet function y t , ( ) they are created.The continuous wavelets can be constructed by [35]: where b is the translation parameter, and a is the dilation parameter.We can restrict a and b to only have discrete values to obtain the corresponding family of discrete wavelets such that = - a a , which results: . 9 m-th order Bernoulli polynomials b t m ( ) are defined as: where a i ( = ¼ i m 1, 2, 3, , ) are Bernoulli numbers.Assuming k, M are positive integers and )are defined as: Chebyshev wavelets are defined similarly.In this study, we employ Chebyshev wavelets of the 3rd kind.Again assuming k, M are positive integers and = 1, discrete Chebyshev wavelets of the 3rd kind on Î t 0, 1 [ )are defined as: where V t m ( ) is m-th order Chebyshev polynomials of the 3rd kind.The recurrence formula for Chebyshev polynomials of the 3rd kind are given as

Function approximation of bernoulli and chebyshev wavelets
If a function y t ( ) is squarely integrable in [0, 1], it can be approximated by using wavelets as: ] and y(t) can be estimated by the finite series such as: where C is a coefficient vector and y t ( ) is the corresponding wavelet vector defined as: The wavelet matrix is defined as , , , where ¢ = - m M 2 k 1 and t i are collocation points.

Operational matrices of bernoulli and chebyshev wavelets
In this section we obtain the fractional operational matrices of Bernoulli and Chebyshev Wavelet.Fractional operational matrices require less computation with Block Pulse Functions (BPFs).Consequently, it is simpler to convert the FDE into a vector-matrix form algebraic equation.All the numerical calculations are performed using Matlab R2021b in this study.
An ¢ m set of BPFs is defined as where = ¢ i m 1, 2, 3, ..., .The functions b t i ( ) are disjoint and orthogonal.For Î t 0, 1 Any squarely integrable function f t ( ) defined in [0, 1) can be expanded into an ¢ m set of BPFs as The wavelet vector can also be expanded to an ¢ m set of BPFs as The Block Pulse operational matrix for fractional integration a F is defined as where The fractional integration of the corresponding wavelet vector y t ( ) defined in (18) can be approximated as: where matrix a ¢´¢ P m m is called the corresponding wavelet operational matrix.Using equations ( 24)-( 27) we obtain and The resulting wavelet operational matrix a ¢ ¢ P m xm for Bernoulli and Chebyshev wavelets becomes For the convergence analysis and the error estimation of the Bernoulli wavelets and Chebyshev Wavelets of the third kind, the reader is referred to [33] and [35], respectively.

Numeric examples
This section includes several FDE electric circuit equation examples solved by BWM and Ch3WM to demonstrate the strength of the method.

Fractional LC circuit
Applying the Wavelet approximation method to the fractional derivative term in equation (1) we obtain: Integrating (31) and incorporating initial conditions result Using (28) and (29) in (1) yields Numerically solving (33) for C coefficients also provides the approximate solution given in equation (16).Equation (1) for the integer order a = 2 gives the solution of classical series LC circuit current with the angular resonance frequency j = LC 1 , The exact and approximate solutions obtained with BWM and Ch3WM for = a = 2 are given in figure 1 in which the exact and approximate solutions agree.The approximate solutions for fractional derivatives a = 1.5, a = 1.7, a = 1.9 and a = 1.99 are provided in figure 2. As can be seen from the figure, as the fractional derivative a approximate 2, the numerical FDE solutions also approximate to the solution obtained for ordinary differential equation.

Fractional RC circuit
Applying the Wavelet approximation method to the fractional derivative term in equation ( 2) we obtain: a = 1 are given in figure 3 in which the exact and approximate solutions agree.The approximate solutions for fractional derivatives a = 0.5, a = 0.7, a = 0.9 and a = 0.99 for both wavelet methods are provided in figure 4. As can be seen from the figure, as the fractional derivative a approximate 1, the numerical FDE solutions also approximate to the solution obtained for ordinary differential equation.

Fractional RLC circuit
Applying the Wavelet approximation method to the fractional derivative term a D q t 2 ( ) in equation ( 3) we obtain: Integrating (37) and incorporating initial conditions results Using (37) and (38) As before, numerically solving (39) for C coefficients also provides the approximate solution given in equation (16).Equation (3) for the integer order a = 1 corresponds to solution of classical series RLC circuit is the square of the angular resonance frequency) parameters, the solution becomes k r = -r q t q e t 0 cos 4 . 0 The exact and approximate solutions obtained with BWM and Ch3WM for 5, in which the exact and approximate solutions are very close.The approximate solutions for fractional derivatives a = 0.7, a = 0.8, a = 0.9 and a = 0.99 for both wavelet methods are provided in figure 6.As can be seen from the figure, as the fractional derivative a approximate 1, the numerical FDE solutions also approximate to the solution obtained for ordinary differential equation.
To be able to compare the both wavelet methods in terms of computational load and speed, we provide FLOPs (Floating Point Operations) numbers for each method for the 3 fractional circuit solutions in tables 1-3.As can be seen from the Tables, BWM and Ch3WM produce very similar FLOPs counts, therefore in terms of computational load, both methods require similar numbers and also the elapsed times for both numerical methods are very close.Table 4 provides the maximum absolute errors for LC and RC FDEs for BWM.Ch3WM provides very close maximum absolute errors therefore only BWM results are included in the table which clearly demonstrates that the larger the m′ , the smaller the error becomes.The tables indicate that the numerical wavelet solution methods are very good approximations for fractional circuit equations.Now we compare our wavelet method solutions with Adams/Bashforth Method [51] for fractional orders.For the LC circuit equation example, we provide a comparison for Bernoulli Wavelet method for the fractional a values of 1.5, 1.7, 1.9 and = v A 0 0.01 ( ) in table 5.As can be seen from the table, the results are very close.Similarly, we provide the solution comparison of Ch3WM with Adams/Bashforth Method for RC circuit equation example for fractional a values of 0.5, 0.7, 0.9 and = u V 0 0.01 ( ) in table 6.Again, we can say that the numeric values agree for both methods.

Conclusions
In this study, we provide numerical solutions for fractional LC, RC, RLC circuit equations.We employ Bernoulli Wavelets and Chebyshev Wavelets of the 3rd kind for the numerical approximations.Chebyshev wavelets are orthogonal, Bernoulli wavelets can be made orthogonal using some special conditions.The orthogonality principal is essential for the fast convergence and low computational load for the wavelet methods.Block pulse functions are used to obtain the required operational matrices, which are used for the transformation from FDEs to systems of algebraic equations.The novel contribution of the BWM and Ch3WM for fractional circuit equations is their fast convergence, low computational load and compactness of the method for easier coding.As the numerical examples demonstrate, the method is fast and very accurate especially for large values of ¢ m .Both methods have similar speed of execution since their FLOPs counts are very similar.What is more, a being a real number does not add extra computational load for the simulations, FLOPs counts are also measured for fractional a values and seen that they are the same as the corresponding integer-order simulations for the same ¢ m .As another comparison, the FDE solutions for fractional a values are also obtained using Adams/Bashforth method and shown to have very similar results as the proposed method solution, which validates the solutions.
Equation (2)for the integer order a = 1 gives the solution of classical series RC circuit voltage

Table 1 .
BWM and Ch3WM FLOPs counts for fractional LC circuit.

Table 2 .
BWM and Ch3WM FLOPs counts for fractional RC circuit.

Table 3 .
BWM and Ch3WM FLOPs counts for fractional RLC circuit.