Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations

Khataybeh, Sana'a Nazmi; Hashim, Ishak; Alshbool, Mohammed

doi:10.1016/j.jksus.2018.05.002

Cited by 24 publications

(14 citation statements)

References 18 publications

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“…The following problem consists of modeling the spreading of a thin oil drop on a horizontal surface. It is the thin-film problem given by Example 4.3 in [16]: [17], the method by Khataybeh et al (M 2 ), and our spline-based method of order m = 9 (M 3 ), for h = 0.01 and b = 1. For the numerical comparison, the exact solution was obtained from Duffy and Wilson [1].…”

Section: A Nonlinear Scalar Problemmentioning

confidence: 99%

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

et al. 2021

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Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.

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Section: A Nonlinear Scalar Problemmentioning

confidence: 99%

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

et al. 2021

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“…[14], khodabin et al [15,16], Zhang [17,18], Janković [19],and Heydari et al [20] employed these functions so that it is possible to solve stochastic Itô-Volterra integral equations, however, other scholars attempted to solve them theoretically or numerically. Since Bernstein polynomials have been known as integrable and differentiable polynomials; therefore these polynomials have been recently utilized to solve integral and differential equations [21,22,23,24]. Employing Bernstein polynomials and their derivatives, approach differential and integral equations are converted to linear or nonlinear algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of linear stochastic Itô-Volterra integral equation using stochastic Bernstein polynomial operational matrices

Sharafi

Basirat

Shahriari

2023

Preprint

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This article has tried to evaluate a computational and practical numerical approach that is on the basis of Bernstein Polynomials for solving linear stochastic Itô-Volterra integral equations. We solve the linear stochastic problem by using the stochastic operational matrix of Bernstein polynomials. First, Bernstein polynomials and their properties are used for deriving a general process in order to make the Bernstein polynomials stochastic operational matrix. Second, using the stochastic operational matrix, the stochastic Itô-Volterra integral equations are solved. Third, the present paper investigates the error analysis of the convergence of the suggested approach. Finally, the accuracy and efficiency of the suggested approach using a numerical example is investigated.

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“…One of the approximation methods consists in simplifying the unknown function based on orthogonal polynomials and operational matrices (OM) so that it becomes a system of algebraic equations that can be easily solved. Many works have explored this method using OM based on some polynomials, for example, the Legendre polynomial [18], the Chebyshev polynomial [19], the Genocchi polynomial [20], the Bernoulli polynomial [21], the Bernstein polynomial [22] and the Wang-Ball polynomial [23].…”

Section: Introductionmentioning

confidence: 99%

The Operational Matrices Methods for Solving Falkner-Skan Equations

Mahdi

AL-Jawary

2022

Iraqi Journal of Science

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The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives a good agreement.

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Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations

Cited by 24 publications

References 18 publications

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

Numerical solution of linear stochastic Itô-Volterra integral equation using stochastic Bernstein polynomial operational matrices

The Operational Matrices Methods for Solving Falkner-Skan Equations

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