A fast and efficient scheme for solving a class of nonlinear Lienard’s equations

Adel, Waleed

doi:10.1007/s40096-020-00328-7

Cited by 14 publications

(2 citation statements)

References 20 publications

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“…Also, Kumar et al [29] used the Rabotnov fractional exponential kernel to solve the nonlinear Lienard equation numerically. More recently, Adel [30] demonstrated an approach based on Bernoulli collocation and shifted Chebyshev collocation points to solve Equations ( 4) and (6).…”

Section: Introductionmentioning

confidence: 99%

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Verma,

Sumelka,

Yadav

2023

Symmetry

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This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods.

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Section: Introductionmentioning

confidence: 99%

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Verma,

Sumelka,

Yadav

2023

Symmetry

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“…In addition to Chebyshev polynomials [13,18], different (orthogonal) polynomial functions have been utilized inside the spectral collocation approaches in the literature. Among others, we mention Dickson [25], Bessel [14,36,48], Legendre [37], Benoulli [2], Vieta-Fibonacci [1], and Jacobi [47], to name a few.…”

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confidence: 99%

Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity

Izadi¹,

Srivástava²

2023

DCDS-S

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Two effective and accurate matrix collocation techniques based on novel generalized shifted Chebyshev functions of the third kind (GSCFTK) are presented to examine the approximate solutions of a fractional-order population model considering the impact of carrying capacity. The fractional operator in the Liouville-Caputo sense is considered. The first direct technique is called the GSCFTK matrix procedure whereas the second one relied on the quasilinearization method and the GSCFTK matrix collocation strategy is called QLM-GSCFTK. In both approaches, the nonlinear population model is transformed into an algebraic system of equations. The resulting matrix equation is nonlinear in the first approach while is linear in the latter one, which is more efficient than the direct matrix method. The convergence analysis of the new generalized bases is established. To testify the robustness and effectiveness of the presented techniques, various numerical simulations are performed using diverse fractional orders. The results prove the applicability of the presented techniques of providing accurate results in comparison to other available numerical models and can be extended to other similar biological problems. Results also show that the numerical schemes are robust.

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A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

Sriwastav,

Barnwal,

Srivastav

et al. 2023

Math Methods in App Sciences

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The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.

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A fast and efficient scheme for solving a class of nonlinear Lienard’s equations

Cited by 14 publications

References 20 publications

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity

A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

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