Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm

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.

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.

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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].

[Figure omitted. See PDF.]

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.

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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.

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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 n^th derivatives with respect to η.

Proposition 2.3 [57] Assume that exists, then

Lemma 2.4 If η^μ 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 P_K(η): when the Laplacian is split in spherical coordinates as functions of the polar angle β with (η = cos β).

* First kind Chebyshev polynomials T_K(η):

* Second kind Chebyshev polynomials U_n(η):

* Gegenbauer polynomials G_K(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 Ξ_n 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 aswhere .

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 Ξ_n(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.

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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 ψ_i 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 D^T 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 η_j, 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 = η_i. 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 [Θ_κ; δ_κ],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 k^th 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 ς_n(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 μ_k(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.

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[Figure omitted. See PDF.]

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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.

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[Figure omitted. See PDF.]

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Example 2: [66] considered that ζ(η) = 1, ξ(η) = 2, ϑ(η) = −1, μ₁(η) = 1 and μ_k(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.

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[Figure omitted. See PDF.]

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Example 3: [33] Consider that ζ(η) = 0, ξ(η) = −1, ϑ(η) = 1, μ₁(η) = 1 and μ_k(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.

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[Figure omitted. See PDF.]

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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.

[Figure omitted. See PDF.]

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.

[Figure omitted. See PDF.]

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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51. 51. El-shenawy A, El-gamel M, Jaheen DR. Numerical Solution of Biharmonic Equation Using Modified Bi-Quintic B-Spline Collocation Method. MEJ Mansoura Engineering Journal. 2022;47(6):14–22.

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52. 52. El-Gamel M, El-Baghdady GI, Abd El-Hady M. Highly efficient method for solving parabolic PDE with nonlocal boundary conditions. Applied Mathematics. 2022;13(2):101–119.

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53. 53. El-Gamel M, Mohamed N, Adel W. Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis. Mathematical Modelling and Numerical Simulation with Applications. 2023;3(4):351–375.

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57. 57. Atangana A. Fractional operators with constant and variable order with application to geo-hydrology. Academic Press; 2017.

58. 58. El-Gamel M, Kashwaa Y, Abd El-Hady M. Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits. Partial Differential Equations in Applied Mathematics. 2024;12:100914.

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60. 60. Singh H. Approximate solution of fractional vibration equation using Jacobi polynomials. Applied Mathematics and Computation. 2018;317:85–100.

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64. 64. Akyüz-Daşcıoğlu A, Çerdi H. The solution of high-order nonlinear ordinary differential equations by Chebyshev series. Applied Mathematics and Computation. 2011;217(12):5658–5666.

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65. 65. Shahmorad S. Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error estimation. Applied Mathematics and Computation. 2005;167(2):1418–1429.

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66. 66. Hosseinnia SH, Ranjbar A, Momani S. Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & Mathematics with Applications. 2008;56(12):3138–3149.

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67. 67. Yüzbaşı Ş. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Applied Mathematics and Computation. 2013;219(11):6328–6343.

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68. 68. Taiwo OA, Osilagun JA. Approximate solution of generalized Riccati differential equations by iterative decomposition algorithm. Int J Eng Innovative Technol. 2012;1(2):53–56.

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69. 69. Sakar MG. Iterative reproducing kernel Hilbert spaces method for Riccati differential equations. Journal of Computational and Applied Mathematics. 2017;309:163–174.

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70. 70. Batiha B, Noorani M, Hashim I. Application of variational iteration method to a general Riccati equation. In: International mathematical forum. vol. 2; 2007. p. 2759–2770.

71. 71. Mabood F, Ismail AIM, Hashim I. Application of optimal homotopy asymptotic method for the approximate solution of Riccati equation. Sains Malaysiana. 2013;4:863–867.

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72. 72. Mehmood A, Zameer A, Aslam MS, Raja MAZ. Design of nature-inspired heuristic paradigm for systems in nonlinear electrical circuits. Neural Computing and Applications. 2020;32:7121–7137.

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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

About the Authors:

Mahmoud Abd El-Hady

Contributed equally to this work with: Mahmoud Abd El-Hady, Mohamed El-Gamel, Homan Emadifar, Atallah El-shenawy

Roles: Data curation, Formal analysis, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing

Affiliation: Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt

Mohamed El-Gamel

Contributed equally to this work with: Mahmoud Abd El-Hady, Mohamed El-Gamel, Homan Emadifar, Atallah El-shenawy

Roles: Formal analysis, Methodology, Project administration, Resources, Supervision, Writing – original draft, Writing – review & editing

Affiliation: Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt

Homan Emadifar

Contributed equally to this work with: Mahmoud Abd El-Hady, Mohamed El-Gamel, Homan Emadifar, Atallah El-shenawy

Roles: Conceptualization, Supervision, Writing – original draft, Writing – review & editing

E-mail: [email protected]

Affiliations: Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, Tamil Nadu, India, MEU Research Unit, Middle East University, Amman, Jordan, Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

ORICD: https://orcid.org/0000-0002-8034-1475

Atallah El-shenawy

Contributed equally to this work with: Mahmoud Abd El-Hady, Mohamed El-Gamel, Homan Emadifar, Atallah El-shenawy

Roles: Conceptualization, Formal analysis, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing

Affiliations: Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, Egypt, Department of Mathematics, Faculty of Sciences, New Mansoura University, New Mansoura, Egypt

ORICD: https://orcid.org/0000-0001-5691-0503

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32. El-Gamel M, Abd-El-Hady M, El-Azab Ms. Chelyshkov-tau approach for solving Bagley-Torvik equation. Applied Mathematics. 2017;8(12):1795.

33. Agheli B. Approximate solution for solving fractional Riccati differential equations via trigonometric basic functions. Transactions of A Razmadze Mathematical Institute. 2018;172(3):299–308.

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36. Geng F. A modified variational iteration method for solving Riccati differential equations. Computers & Mathematics with Applications. 2010;60(7):1868–1872.

37. Khan NA, Ara A, Jamil M. An efficient approach for solving the Riccati equation with fractional orders. Computers & Mathematics with Applications. 2011;61(9):2683–2689.

38. Merdan M. On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative. International Journal of differential equations. 2012;2012(1):346089.

39. Sakar MG, Akgül A, Baleanu D. On solutions of fractional Riccati differential equations. Advances in Difference Equations. 2017;2017:1–10.

40. Ghomanjani F, Khorram E. Approximate solution for quadratic Riccati differential equation. Journal of Taibah university for science. 2017;11(2):246–250.

41. Ezz-Eldien SS, Machado JAT, Wang Y, Aldraiweesh AA. An algorithm for the approximate solution of the fractional Riccati differential equation. International Journal of Nonlinear Sciences and Numerical Simulation. 2019;20(6):661–674.

42. Xin Liu K, Yao Y. Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative. Journal of Mathematics. 2020;2020(1):1274251.

43. bin Rasedee AFN, Sathar MHA, Ishak N, Hamzah SR, Jamaludin NA. Numerical Approximation of Riccati type differential equations. ASM Science Journal. 2020;.

44. Singh J, Gupta A, Kumar D. Computational Analysis of the Fractional Riccati Differential Equation with Prabhakar-type Memory. Mathematics. 2023;11(3):644.

45. El-Gamel M, El-Shamy N. B-spline and singular higher-order boundary value problems. SeMA Journal. 2016;73:287–307.

46. El-Gamel M, Abd El-Hady M. Numerical solution of the Bagley-Torvik equation by Legendre-collocation method. SeMA Journal. 2017;74:371–383.

47. El-Gamel M, El-Shenawy A. A numerical solution of Blasius equation on a semi-infinity flat plate. SeMA Journal. 2018;75:475–484.

48. El-Gamel M, Abdrabou A. Sinc-Galerkin solution to eighth-order boundary value problems. SeMA Journal. 2019;76:249–270.

49. El-Gamel M, Mohamed O, El-Shamy N. A robust and effective method for solving two-point BVP in modelling viscoelastic flows. Applied Mathematics. 2020;11(01):23.

50. El-Gamel M, Abd El-Hady M. On Using Bernstein Scheme for Computation of the Eigenvalues of Fourth-Order Sturm–Liouville Problems. International Journal of Applied and Computational Mathematics. 2021;7:1–18.

51. El-shenawy A, El-gamel M, Jaheen DR. Numerical Solution of Biharmonic Equation Using Modified Bi-Quintic B-Spline Collocation Method. MEJ Mansoura Engineering Journal. 2022;47(6):14–22.

52. El-Gamel M, El-Baghdady GI, Abd El-Hady M. Highly efficient method for solving parabolic PDE with nonlocal boundary conditions. Applied Mathematics. 2022;13(2):101–119.

53. El-Gamel M, Mohamed N, Adel W. Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis. Mathematical Modelling and Numerical Simulation with Applications. 2023;3(4):351–375.

54. El-shenawy A, gamel M E, Anany ME. A Novel Scheme Based on Bessel Operational Matrices for Solving a Class of Nonlinear Systems of Differential Equations. MEJ Mansoura Engineering Journal. 2024;49:1–10.

55. El-shenawy A, El-Gamel M, Reda D. Troesch’s problem: A numerical study with cubic trigonometric B-spline method. Partial Differential Equations in Applied Mathematics. 2024; p. 100694.

56. El-Shenawy A, El-Gamel M, Abd El-Hady M. On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application. Heat Transfer. 2024; p. 1–29.

57. Atangana A. Fractional operators with constant and variable order with application to geo-hydrology. Academic Press; 2017.

58. El-Gamel M, Kashwaa Y, Abd El-Hady M. Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits. Partial Differential Equations in Applied Mathematics. 2024;12:100914.

59. Van Assche W. Ordinary Special Functions. In: Encyclopedia of Mathematical Physics. Oxford: Academic Press; 2006. p. 637–645.

60. Singh H. Approximate solution of fractional vibration equation using Jacobi polynomials. Applied Mathematics and Computation. 2018;317:85–100.

61. Behroozifar M, Sazmand A. An approximate solution based on Jacobi polynomials for time-fractional convection–diffusion equation. Applied Mathematics and Computation. 2017;296:1–17.

62. Doha EH, Bhrawy AH, Ezz-Eldien SS. A new Jacobi operational matrix: an application for solving fractional differential equations. Applied Mathematical Modelling. 2012;36(10):4931–4943.

63. Abd El-Hady M, El-shenawy A. Jacobi polynomials and the numerical solution of ray tracing through the crystalline lens. Optical and Quantum Electronics. 2024;56(8):1329.

64. Akyüz-Daşcıoğlu A, Çerdi H. The solution of high-order nonlinear ordinary differential equations by Chebyshev series. Applied Mathematics and Computation. 2011;217(12):5658–5666.

65. Shahmorad S. Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error estimation. Applied Mathematics and Computation. 2005;167(2):1418–1429.

66. Hosseinnia SH, Ranjbar A, Momani S. Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & Mathematics with Applications. 2008;56(12):3138–3149.

67. Yüzbaşı Ş. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Applied Mathematics and Computation. 2013;219(11):6328–6343.

68. Taiwo OA, Osilagun JA. Approximate solution of generalized Riccati differential equations by iterative decomposition algorithm. Int J Eng Innovative Technol. 2012;1(2):53–56.

69. Sakar MG. Iterative reproducing kernel Hilbert spaces method for Riccati differential equations. Journal of Computational and Applied Mathematics. 2017;309:163–174.

70. Batiha B, Noorani M, Hashim I. Application of variational iteration method to a general Riccati equation. In: International mathematical forum. vol. 2; 2007. p. 2759–2770.

71. Mabood F, Ismail AIM, Hashim I. Application of optimal homotopy asymptotic method for the approximate solution of Riccati equation. Sains Malaysiana. 2013;4:863–867.

72. Mehmood A, Zameer A, Aslam MS, Raja MAZ. Design of nature-inspired heuristic paradigm for systems in nonlinear electrical circuits. Neural Computing and Applications. 2020;32:7121–7137.