https://orcid.org/0000-0002-8034-1475
Figures
Abstract
This paper focuses on modeling Resistor-Inductor (RL) electric circuits using a fractional Riccati initial value problem (IVP) framework. Conventional models frequently neglect the complex dynamics and memory effects intrinsic to actual RL circuits. This study aims to develop a more precise representation using a fractional-order Riccati model. We present a Jacobi collocation method combined with the Jacobi-Newton algorithm to address the fractional Riccati initial value problem. This numerical method utilizes the characteristics of Jacobi polynomials to accurately approximate solutions to the nonlinear fractional differential equation. We obtain the requisite Jacobi operational matrices for the discretization of fractional derivatives, therefore converting the initial value problem into a system of algebraic equations. The convergence and precision of the proposed algorithm are meticulously evaluated by error and residual analysis. The theoretical findings demonstrate that the method attains high-order convergence rates, dependent on suitable criteria related to the fractional-order parameters and the solution’s smoothness. This study not only improves comprehension of RL circuit dynamics but also offers a solid numerical foundation for addressing intricate fractional differential equations.
Citation: Abd El-Hady M, El-Gamel M, Emadifar H, El-shenawy A (2025) Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm. PLoS ONE 20(1): e0316348. https://doi.org/10.1371/journal.pone.0316348
Editor: António M. Lopes, University of Porto Faculty of Engineering: Universidade do Porto Faculdade de Engenharia, PORTUGAL
Received: August 5, 2024; Accepted: December 5, 2024; Published: January 14, 2025
Copyright: © 2025 Abd El-Hady et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: No data sets were generated or analyzed during this study. The tables and graphs in this article are based on the generated results in the tables with the help of Matlab.
Funding: Authors do not receive any fund.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Recently, non-integer derivatives are used in ordinary differential equations to model various real-world events, for instance, fluid dynamics, visco-elastic damping, fluid flow tracing, electro-chemistry, electrical circuits, physics, the model of neurons in biology, voltage dividers, quantum mechanics, electromagnetism, hydrology, mathematical biology, etc [1–4]. Furthermore, fractional derivatives are widely renowned for their ability to efficiently govern models [5–10]. Recent developments in fractional calculus have broadened its applications in numerous scientific and engineering domains. A notable advancement is the introduction of novel fractional operators, like the Atangana-Baleanu fractional derivative [11], which enhance modeling efficacy for intricate systems exhibiting memory effects and non-local characteristics. Moreover, academics have devised advanced numerical approaches, such as spectral methods and fractional finite element methods, to effectively address fractional differential equations. These improvements have resulted in substantial breakthroughs in control systems, signal processing, image processing, and biotechnology. Fractional-order controllers have demonstrated the ability to boost the performance of dynamical systems, whilst fractional-order filters have enhanced noise reduction and signal denoising.
Mathematicians and physicists have worked hard to create strong numerical and analytical methodologies for obtaining the solutions of fractional differential equations. Numerical approaches have been frequently employed in the last two decades to get approximate solutions to applied and scientific problems such as finite differences, finite elements, boundary element techniques, etc. A summary of the essential techniques related to the approximate solution of various differential equations with fractional derivatives is displayed in Table 1.
Fractional-order circuit components, such as capacitors and inductors, have grown in popularity because they can better simulate frequency-dependent behavior than standard integer-order models. The fractional-order RL circuit, in particular, has been applied in a variety of domains, including power electronics and control systems. However, the addition of a nonlinear inductor complicates the mathematical description of the circuit, necessitating the development of specific numerical approaches. One such approach that has been deployed is the Jacobi collocation method JCM. Fig 1 illustrates the fractional-order resistor-inductor (RL) electric circuit with a non-linear inductor, as stated in the references [12, 13]. The approach of nonlocal Riccati, resistor-inductor circuit problem was investigated by applying the integro spline quasi-interpolation technique in [14].
In this work, we aim to study one of the famous problems in electricity and electrical engineering: the Riccati equation. The design and analysis of the RL circuit shown in the Fig 1 are modeled by the nonlinear initial value Riccati problem with fractional derivative.
Firstly, let us define the general form of the Riccati equation with fractional derivative as follows:
(1.1)
with
(1.2)
In the realm of RL circuits, fractional-order elements can be understood as generalized inductors or resistors exhibiting memory effects. A fractional-order inductor can be seen as a device that not only resists alterations in current but also retains a memory of previous current values. A fractional-order resistor is a component whose resistance is contingent upon the historical voltage applied across it.
Motivation for Fractional-Order Modeling
- Memory effects: Real-world components frequently demonstrate memory effects, wherein their current behavior is influenced by prior situations. Fractional-order models can proficiently encapsulate these memory effects, resulting in more precise descriptions of circuit dynamics.
- Non-perfect components: Conventional models frequently presume perfect components, which may not consistently hold true in practical situations. Fractional-order models can include non-ideal phenomena, such as frequency-dependent resistance or inductance, offering a more accurate representation of circuit behavior.
- Complex dynamics: Fractional-order differential equations can demonstrate a broader spectrum of dynamic behaviors compared to their integer-order equivalents. This enhanced flexibility facilitates more precise modeling of intricate systems, especially those exhibiting chaotic or oscillatory behavior.
Fractional Riccati equations are commonly encountered in diverse fields such as applied sciences, engineering, and real-world applications. These equations are relevant in areas such as diffusion problems, control theory, pattern generation in dynamic gas systems, network synthesis, river flows, invariant embedding, and econometric models [33]. Because of the Riccati differential equation’s relevance and numerous applications, scientists have done countless investigations to find more efficient and precise methods for solving it. Several analytical and numerical techniques have arisen in the literature to tackle this problem. A short survey considering these approaches that have been proposed in Table 2.
The collocation technique offers an extremely efficient spectral approach that may be flexibly utilized to estimate the approximate solution of ordinary and partial differential equations at predefined points in the solution domain. The unknown function is represented as a truncated series of basis functions, with coefficients determined by solving equations. The type of equations to be solved depends on the characteristics of the initial problem and the collocation methods utilized. El-Gamel and his research group have widely contributed to this field over the last ten years. They developed novel strategies and techniques based on various basis functions to approximate the solutions of differential, integral, and integro-differential equations in engineering applications. A brief timeline of the main papers is arranged in the diagram shown in Fig 2.
This paper contains multiple original contributions to the analysis and resolution of RL electric circuits represented by fractional Riccati initial value problems (IVPs). We provide a novel implementation of the fractional Riccati model to appropriately represent the intricate memory-dependent behavior of RL circuits, marking a substantial improvement over conventional integer-order models that frequently inadequately characterize these systems. Additionally, we offer a novel numerical technique that integrates the Jacobi collocation method with a Broyden-modified Newton algorithm. This hybrid Jacobi-Broyden Newton method facilitates the efficient and precise solution of the nonlinear fractional differential equation by utilizing the operational matrices of Jacobi polynomials. This integration improves computing efficiency and guarantees superior convergence rates, as rigorously evidenced by error and residual assessments. This study enhances existing numerical approaches and applies them to a more realistic fractional model, surpassing previous research in both theoretical and practical dimensions of resolving RL circuit issues, hence paving the way for further exploration in fractional-order electrical systems. This paper offers multiple original contributions to the analysis and resolution of RL electric circuits represented by fractional Riccati initial value problems (IVPs). We provide a novel use of the fractional Riccati model to effectively capture the intricate memory-dependent behavior of RL circuits, representing a substantial improvement over conventional integer-order models that frequently inadequately characterize these systems. Additionally, we offer a novel numerical technique that integrates the Jacobi collocation method with a Broyden-modified Newton algorithm. This hybrid Jacobi-Broyden Newton method facilitates the efficient and precise solution of the nonlinear fractional differential equation by utilizing the operational matrices of Jacobi polynomials. This integration improves computing efficiency and guarantees superior convergence rates, as rigorously evidenced by error and residual assessments. This study enhances existing numerical approaches and applies them to a more realistic fractional model, surpassing previous research in both theoretical and practical dimensions of solving RL circuit issues, hence paving the way for further exploration in fractional-order electrical systems.
The organization of this manuscript is structured as follows: Some preliminary material contains the basic definitions, lemmas, and theorems related to fractional derivatives, and the basic steps of Jacobi’s polynomials and their operational matrix are indeed accessible in the second section. Section 3 deals with the derivation of the discrete system obtained from the Jacobi collocation method. Section 4 derives the residual error of the proposed technique. Section 5 reveals numerical simulations and comparisons with other methods. Finally, the conclusions are given in Section 6.
2 Preliminaries and relations
In the following section, the basic theorems and relations that will be utilized in the paper will be considered.
2.1 Fractional derivative
Definition 2.1 The Caputo fractional integral
for the real function Ξ(η) is represented by [6]
Definition 2.2 The fractional derivative operator of Caputo type
for function Ξ(η) with order α > 0 is defined by [6]
where
-
; Gamma function,
- Ξ(n)(η) is the classical nth derivatives with respect to η.
Proposition 2.3 [57] Assume that exists, then
Lemma 2.4 If ημ is a polynomial of μ− degree and , then
where α, μ ≥ 0 and ⌈α⌉ is ceiling function.
2.2 An overview of Jacobi polynomials
Definition 2.5 Jacobi polynomial
and η ∈ [−1, 1] is given by
Jacobi polynomials have the following properties [58].
- Orthogonal polynomials over the interval η ∈ [−1, 1] with respect to the weight w(η).
(2.1)
- The Rodrigues formula
-
,
.
-
.
θ1 and θ2 significantly influence the characteristics of the Jacobi polynomials. Here are some special cases of Jacobi polynomials based on the values of θ1 and θ2 [59].
- Legendre polynomials PK(η): when the Laplacian is split in spherical coordinates as functions of the polar angle β with (η = cos β).
- First kind Chebyshev polynomials TK(η):
- Second kind Chebyshev polynomials Un(η):
- Gegenbauer polynomials GK(η) or ultraspherical polynomials are Jacobi polynomials with equal parameters.
For more flexibility, the Jacobi polynomials are redefined over the interval [0, 1]; they will be denoted by shifted Jacobi polynomials.
Definition 2.6 The shifted Jacobi polynomial is a polynomial of degree i defined over the range η ∈ [0, 1] and is defined by [60]
(2.2)
2.3 Convergence of Jacobi polynomial
In this part, we examine the convergence and error bounds for Jacobi polynomials. The following three lemmas establish the upper limit of approximation error, as demonstrated in [61]. To derive an upper bound on approximation error, assume the function Ξ has the first n + 1 continuous derivatives, i.e. Ξ ∈ Cn+1[0, 1].
Lemma 2.7 Suppose that the function
has n + 1 continuous derivatives, Ξ ∈ Cn+1[0, 1] and Ξn is the best approximation of Ξ in the space
where
then
where M = max|Ξ(n+1)(η)| over 0 ≤ η ≤ 1.
Lemma 2.7 demonstrates that the optimal approximation approaches Ξ as the derivative Ξ satisfies the continuity requirement, as n tends towards infinity.
Definition 2.8 The function Ξ on [0, 1] has modulus of continuity which is defined as
where
.
Lemma 2.9 Ξ(η) is uniformly continuous on [0, 1] iff .
Lemma 2.10 Suppose that Ξ(η) is bounded on [0, 1] and Ξ in the space then
we have
, then
Ξn(η) = Ξ(η).
3 Jacobi-Collocation scheme
Jacobi Collocation Method (JCM) is developed to obtain an approximate solution for the nonlinear fractional Riccati problem Eq (1.1), along with the given initial conditions (1.2). The full algorithm is divided into three stages and explained step by step in Fig 3.
To apply the collocation scheme for calculating in the solution of (1.1), the function Ξ(η) is approximated in terms of a truncated shifted Jacobi polynomial series as follows:
(3.1)
where
are the orthogonal shifted Jacobi polynomials and
are unknown coefficients which can be calculated via the following form:
In the above formula, w(η) are the weights given by Eq (2.1) and ψi are represented in the following expanded form for simplicity:
Eq (3.1) is written as follows
(3.2)
where
(3.3)
Lemma 3.1 [62] If the shifted Jacobi polynomial is defined in (2.2) then
(3.4)
And the derivatives in (3.4) can be written as
(3.5)
such that DT is a square matrix has (n + 1) rows and (n + 1) columns and can be written as follows:
where [63]
Lemma 3.2 [62] If defined in (2.2) is the shifted Jacobi polynomial then
(3.6)
Lemma 3.3 [62] Let be shifted Jacobi polynomials defined in (2.2) then
(3.7)
where operational matrix of fractional derivative is D(α) with dimensions (n + 1) × (n + 1)
(3.8)
and
and
(3.9)
Lemma 3.4 [64]
(3.10)
where
and
By substitute Eqs (3.1) in (1.1) we obtian
(3.11)
Now, we may substitute the collocation points ηj, where j = 0, 1, …, n, with the roots of in Eq (3.11). As a result, we obtain
(3.12)
The next theorem will be obtained:
Theorem 3.5 If (1.1) is the estimated solution of the fractional Riccati differential equations (3.1), then the problem is reduced to solve the following nonlinear system for calculating the unknown coefficients.
(3.13) where
and
proof: Each term in Eq (1.1) is replaced with its corresponding approximation from Eqs (3.2), (3.5), (3.7), and (3.10), while substituting η = ηi. The initial conditions are obtained from Eq (1.2) and will be represented using matrices.
(3.14)
Moreover, the matrix form of the initial conditions becomes
(3.15)
where
Consequently, in the augmented matrix [Q; G] if we replace the ⌈mα⌉ rows by the row matrices [Θκ; δκ],then the final system is obtained
(3.16)
The unknown coefficients in the above nonlinear system of equations can be determined by utilizing the Quasi-Newton Broyden’s approach. The main steps of the scheme are summarized in Algorithm 1.
Algorithm 1: JCM scheme of solution
• Input n, θ1 and θ2.
• Define the n + 1 collocation points over the interval [a, b].
• Approximate the solution Ξ(η) via truncated Lucas series:
• Substitute the approximate series representation of Ξ(η) into BVP.
• Apply the collocation points to the differential equation to obtain a discrete system.
• Incorporate the boundary equations into the discrete system.
• Use the quasi-Newton Broyden’s algorithm to solve the obtained nonlinear system for the unknowns .
3.1 Quasi-Newton Broyden’s algorithm
The quasi-Newton Broyden’s method uses the traditional iterative technique with some modifications to speed up the execution time to reach the equilibrium point. At the beginning, the system of nonlinear (n + 1) equations will be written in the following form:
(3.17)
where
- Π is the unknowns’ vector.
- ω is the equations’ vector.
Let Π(k) represent the estimated values of the unknown coefficients for the kth iteration. Let ω(k) denote the value of ω at the kth iteration. Given that the magnitude of ‖ω(k)‖ is sufficiently enough, we seek to update the vectors ΔΠ(k).
(3.18)
such that the vector ω evaluated at Π(k+1) is equal to zero. Applying the multidimensional extension of Taylor expansion to approximate the change in ω(Υ) near Π yields:
(3.19)
where ω′(Π(k)) is the system’s Jacobian matrix.
(3.20)
If the higher order terms and neglected and by assuming that, J(k) is the Jacobian at Π(k) then Eq (3.19) will be reduced to:
(3.21)
By enforcing ω(Π(k) + ΔΠ(k)) toward zero, then we have:
(3.22)
It is unnecessary to compute the value of J at each iteration step due to the excessive time it consumes. Instead, the Jacobian will be revised by employing the subsequent formula:
If the absolute difference between two successive iterations is less than a user-defined tolerance, i.e., when ∥Π(k+1) − Π(k)∥ ≤ ε, then the process will be terminated.
Algorithm 2: Quasi-Newton Broyden’s
• Give an initial Π = Π(0).
• Calculate J(0), ω(0), Π(1).
• For k = 1, 2, 3, …
• Find ω(k), Δω(k), ΔΠ(k).
• If ‖ω(k)‖ is small enough, stop.
• Calculate .
• Solve J(k) ΔΠ(k) = −ω(k).
• Compute Π(k+1) = Π(k) + ΔΠ(k).
• End.
4 Residual error estimation
The residual error function was used to estimate errors in the Jacobi-collocation approach. [65]. Consider the residual function ςn(η) is:
(4.1)
where
is the JCM approximate solution (3.1) for Eqs (1.1) and (1.2). The following equation will be obtained if we subtract the Eqs (4.1) from (1.1).
(4.2)
Then the linear and nonlinear terms are manipulated, respectively as follows:
such that ε(η) is the function of error and Ξ(η) is the exact solution of Eq (1.1),
(4.3)
(4.4)
Eq (4.2) can be written in the following operator form:
(4.5)
Consequently,
(4.6)
the above equation has the following solution:
By applying the proposed algorithm in Section 3, the unknown values are obtained. The maximum absolute errors will be calculated by:
(4.7)
Through the utilization of maximum error estimate, we may assess the dependability of the outcomes, particularly when an exact solution is not available.
5 Results and simulations
In this section, we first provide three test problems to test the JCM for solving the Riccati problem with various nonlinear terms. Furthermore, we compare our methods, namely JCM at θ1 = θ2 = 0 in all examples with other methods introduced in previous works [18, 33, 66, 67]. In the second part, the application of the RL problem is illustrated and solved with JCM to test the proposed algorithm and showcase its accuracy and reliability.
5.1 Test problems
Example 1 [66] Assume problem (1.1) with its parameters as follows ζ(η) = 1, ξ(η) = 0, ϑ(η) = −1, μ1(η) = 1 and μk(η) = 0 for k = 2, 3, ⋯, m.
For the sake of calculating error bounds, we use the exact solution at α = 1 in the form:
In Tables 3–5, a comprehensive comparison is presented for the numerical solution obtained using the following techniques; Joint Collocation Method (JCM) in conjunction with the improved Adams-Bashforth-Moulton method (IABMM) [66], the modified homotopy perturbation method (MHPM) [18], the enhanced homotopy perturbation method (EHPM) [66] and the Bernstein collocation method (BCM) [67]. Table 3 denotes the values of the solutions for α = 0.75; Table 4 shows them for α = 0.9; and Table 5 gives the values of the solutions for α = 1.
The absolute error for the case α = 1 obtained by our proposed method JCM is compared to the absolute errors derived from the variational iteration method (VIM), the iterative decomposition algorithm (IDA), and the iterative reproducing kernel Hilbert spaces method (IRKHSM) in Table 6. The numerical values demonstrate the superior precision of JCM compared to the other methods. Fig 4 illustrates the estimated solutions for n = 11, α = 0.75, 0.9, 1., specifically for example 1. However, the comprehensive outcomes are graphically represented as bar charts and may be found in Fig 5.
Example 2: [66] considered that ζ(η) = 1, ξ(η) = 2, ϑ(η) = −1, μ1(η) = 1 and μk(η) = 0 for k = 2, 3, …, m in problem (1.1), then
The exact solution of the this problem for α = 1 is
The numerical results found in Table 7 are being compared with different approaches. [18, 66, 67] for α = 1. TThe comparison examines well-known methodologies: variational iteration method (VIM), modified homotopy perturbation method (MHPM), Optimal homotopy asymptotic method (OHAM), and iterative reproducing kernel Hilbert spaces method (IRKHSM). The current method’s absolute errors for α = 1 are compared with the absolute errors of other procedures in Table 8. The comparison of the approximate solution at certain places and the error from the JCM and other approaches are represented by Fig 6. The results are also presented in Tables 7 and 8. Fig 7 shows the JCM’s estimated solution for n = 11 at different values of the fractional derivative’s order: α = 0.75, 0.9, 1. for Example 2. Based on the data acquired, it is evident that the results produced by our suggested approach are more precise in comparison to the other outcomes.
Example 3: [33] Consider that ζ(η) = 0, ξ(η) = −1, ϑ(η) = 1, μ1(η) = 1 and μk(η) = 0 for k = 2, 3, …, m, in problem (1.1), then we obtain:
The exact solution for α = 1 of this problem is
The Table 9 presents a comparison between the numerical solutions obtained using our approach and the real absolute errors for γ = 1. The solutions and errors are also compared with those obtained using trigonometric basic functions (TBF). The current absolute errors for α = 1 are compared with the estimated absolute errors in Table 10. All these data appear in Tables 9 and 10 are displayed in Fig 8 for the comparisons of approximate solution and the absolute error. The results exhibit a clear pattern of consistency, with a significant disparity between the inaccuracy of JCM and TBF. Fig 9a illustrates the estimated solutions for n = 11, α = 0.75, 0.9, 1., specifically for the case of 3. The Fig 9b illustrates the contrast between the actual absolute error and the estimated absolute error functions.
5.2 Analysis of LR circuit
Consider the RL circuit shown in Fig 1. The circuit with second-order non-linearity can be modeled by an initial value problem (IVP) [12, 72], as follows:
(5.1)
subject to initial conditions
(5.2)
In Eq (5.1), The terms Ξ and σ respectively denote flux linkage in the inductor and the induction parameter. The non-linearity of the nonlinear inductor can be mathematically stated as:
(5.3)
where i(Ξ) represents the current. For the transient analysis of the circuit shown in Fig 9, Kirchhoff’s second law is applied, which results in the following equation:
(5.4)
JCM is used in a variety of circumstances where variable resistance, current, voltage, and inductance values are considered in nonlinear electrical circuit models. Assume the integer derivative order is α = 1.
In scenario 1, a constant voltage magnitude of U(η) = 200V and inductance σ = 1 are utilized, while the resistance R is adjusted. The values of resistance, denoted by R, are 100 ohms, 125 ohms, and 155 ohms for instances 1, 2, and 3, respectively. The resistance R is varied while keeping voltage and inductance constant, we observe that higher resistance values lead to a decrease in flux linkage. This is consistent with physical intuition since increasing resistance limits current flow, subsequently reducing the magnetic flux produced in the inductor.
In scenario 2, the analysis focuses on the changes in voltage U(η), while maintaining a constant resistance value of 100Ω and an inductance parameter of σ = 1. Three scenarios are examined using U(η) = 150V, 200V, and 250V for instances 1, 2, and 3, respectively. This case shows that if U(η) is variable with constant resistance and inductance, the results show that increasing voltage leads to a proportionate increase in flux linkage. This behavior is expected because higher input voltage generally drives higher current through the circuit, resulting in greater flux linkage.
In scenario 3 of the nonlinear RL circuits issue, the voltage U(η) is set to 200V and the resistance R is set to 100Ω, while the change in the inductance parameter σ is taken into account. Three examples are analyzed using standard deviations of 0.5, 1, and 1.5 for cases 1, 2, and 3, respectively. This scenario involves varying inductance while holding voltage and resistance constant. The results indicate that larger inductance values yield higher flux linkage. This makes sense as the inductor’s ability to store magnetic energy is directly proportional to its inductance.
In scenario 4, a constant voltage magnitude of U(η) = 20 V and an inductance parameter of σ = 0.5 are utilized, while the resistance R is adjusted. The values of resistors, namely R = 1KΩ, 2KΩ, and 3KΩ, are being taken into account. This scenario examine a case with a high resistance R and a lower constant voltage, with variable resistance values. The results illustrate that even with smaller changes in resistance, flux linkage decreases as resistance increases. This demonstrates the circuit’s sensitivity to resistance when high resistance values are present, which may be particularly relevant for certain practical applications involving power dissipation.
Fig 10 depicts the approximate flux linkage of the inductor using JCM, which coincides with the results using the Runge-Kutta method for scenarios 1, 2, 3 and 4.
Fig 11 illustrates the behavior of the approximate flux linkage of the inductors at various values of the fractional parameter α = 0.2, 0.5, 1 for the RL circuit, while the values of R = 100Ω, the inductance parameter σ = 0.5 and the voltage U(η) = 200V are kept fixed. This figure illustrates the behavior of the approximate flux linkage of the inductor for different values of α. The observations are as follows:
- α = 0.2: The flux linkage rises sharply initially but then stabilizes at a lower value. This behavior suggests a system with a faster initial response and slower decay.
- α = 0.5: The curve shows a moderate rise and a slower decay compared to α = 0.2. This indicates a system with a balance between the initial response and decay.
- α = 1: The curve represents the standard integer-order RL circuit. It exhibits a relatively slower initial rise and faster decay compared to the fractional-order cases.
6 Conclusions and future work
This study introduces the shifted Jacobi-collocation approach as a means to get approximate solutions for fractional Riccati-type differential equations, which belong to the category of fractional differential equations. This technique converts the fractional Riccati-type differential problem into a set of nonlinear algebraic equations. The approximate answers can be derived by solving the resultant system. An appreciable benefit of this approach is that the answers may be readily acquired using computer tools like as MATLAB, MAPLE, and MATHEMATICA. The technique’s validity and usefulness are demonstrated through the inclusion of numerical examples. The aforementioned examples were executed on a computer with code written in MATLAB. The exhibited cases provide favorable outcomes, showcasing the efficacy of this strategy in achieving a high level of concordance between the approximate and precise values. The technique was utilized to investigate the relationship between flux and the variations in induction, resistance, and voltage values in the RL electric circuit. In summary, the Jacobi-collocation method is a highly appealing technique for solving fractional Riccati-type differential equations. It is capable of producing precise approximation solutions with the use of easily accessible computing resources. The RL circuit model derived from the Riccati equation can be generalized for various RL circuits, especially when incorporating fractional-order calculus. This includes the integration of nonlinear elements, fractional-order dynamics, and diverse boundary and initial conditions, such as circuits with time-varying voltage sources or initial magnetic fluxes, which can be accurately modeled by customizing these conditions. In conclusion, by altering parameters, boundary conditions, or incorporating supplementary terms, the Riccati equation framework employed herein can be effectively modified to model various RL and RLC circuits, serving as a robust instrument for the analysis of diverse nonlinear and fractional-order electrical systems. The RL circuit model derived from the Riccati equation can be generalized for various RL circuits, especially when incorporating fractional-order calculus. This includes the integration of nonlinear elements, fractional-order dynamics, and diverse boundary and initial conditions; for instance, circuits with time-varying voltage sources or initial magnetic fluxes can be accurately modeled by customizing these conditions. In conclusion, by altering parameters, boundary conditions, or incorporating supplementary terms, the Riccati equation framework employed here can be effectively modified to model various RL and RLC circuits, serving as a robust instrument for the analysis of diverse nonlinear and fractional-order electrical systems.
Acknowledgments
The authors would like to thank the academic editor and the anonymous referees for their helpful comments and suggestions which improved the original version of this manuscript.
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