Lucas polynomial solution of nonlinear differential equations with variable delays

This paper presents the Lucas polynomial solution of second-order nonlinear ordinary differential equations with mixed conditions. Lucas matrix method is based on collocation points together with truncated Lucas series. The main advantage of the method is that it has a simple structure to deal with the nonlinear algebraic system obtained from matrix relations. The method is applied to four problems. In the first two problems, exact solutions are obtained. The last two problems, Bratu and Duffing equations are solved numerically; the results are compared with the exact solutions and some other numerical solutions. It is observed that the application of the method results in either the exact or accurate numerical solutions.

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“…Our aim is to seek the approximate Lucas solution of Eq. 1, which can be expressed in matrix form (see [8,9])…”

Section: Fundamentals Of the Numerical Methodsmentioning

confidence: 99%

Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

Gümgüm

Baykuş-Savaşaneril

Kürkçü

et al. 2020

Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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“…Another important problem is the Duffing equation which is used in the modeling of many physical phenomena including classical oscillator in chaotic systems, orbit extraction, the prediction of diseases and nonlinear vibration of beams and plates. Hence, Duffing equation has been solved by using many numerical methods such as the Laplace decomposition algorithm [51], shifted Chebyshev polynomials [3], Runge-Kutta-Fehlenberg algorithm [23], Daftardar-Jafari method [1], Adomian decomposition method [49], differential transform method [44], the improved Taylor matrix technique [8], generalized differential quadrature method [30], Legendre wavelets [36], homotopy perturbation method [31], Lucas polynomial approach [22], cubination method [5] and iterative splitting method [29].…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of Troesch and Duffing equations by Taylor wavelets

Özaltun

Gümgüm

2023

Hacettepe Journal of Mathematics and Statistics

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The aim of this study is obtain accurate numerical results for the Troesch and Duffing equations by using Taylor wavelets. Orthonormality property of these polynomials is an advantage of the method since it reduces the computational cost. These equations are known to be stiff equations, thus most of the numerical methods approximates the nonlinear terms and the force function while wavelets do not require such an approximation which is another advantage of the method. Troesch and Duffing equations are solved for several parameters to show the accuracy and efficiency of the applied method. Results show that the proposed method is highly efficient in the solution of indicated type of problems by using quite low degree polynomials.

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“…In addition to these, pantograph equations have been solved using shifted orthonormal Bernstein polynomials [19]. There are also studies interested in solving a nonlinear version of equation (1.2); the reader can see [20], for example, where Lucas polynomials are employed in conjunction with collocation points to solve nonlinear delay differential equations of this type.…”

Section: Introductionmentioning

confidence: 99%

A Galerkin-like method for solving linear functional differential equations underinitial conditions

Yüzbaşı¹,

Karaçayır²

2020

Turk J Math

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In this paper, we present a weighted residual Galerkin method to solve linear functional differential equations. We consider the problem with variable coefficients under initial conditions. Assuming the exact solution of the problem has a Taylor series expansion convergent in the relevant domain, we seek a solution of the given problem in the form of a polynomial having degree N of our choice. Substituting this polynomial with unknown coefficients in the given equation yields an expression linear in these coefficients. We then proceed as in the weighted residual method and take inner product of this expression with monomials up to degree N , resulting in N + 1 linear algebraic equations. Appropriately incorporating the initial conditions and solving the resulting linear system, we obtain the approximate solution to the given problem. Additionally, we present a way of estimating the absolute error of the obtained approximation, which is then used to improve the original approximation through a method called residual correction. We also show that the upper bound for the error of the proposed method depends on the Taylor truncation error of the exact solution. The proposed scheme and the residual correction technique are illustrated in several example problems.

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Lucas polynomial solution of nonlinear differential equations with variable delays

Cited by 22 publications

References 39 publications

Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

Numerical solutions of Troesch and Duffing equations by Taylor wavelets

A Galerkin-like method for solving linear functional differential equations underinitial conditions

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