Legendre-coefficients Comparison Methods for the Numerical Solution of a Class of Ordinary Differential Equations

Olagunju, Adeyemi Sunday

doi:10.9790/5728-0221419

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…It is to be noted that the expression "Legendre polynomial" when no other qualification is attached exclusively refers to the Legendre polynomials ) (x P n of the first kind. As established in literatures (see ref [8], [10]) Legendre polynomial are special cases of the Legendre function and they satisfy the Legendre equation defined as:…”

Section: Legendre Polynomialsmentioning

confidence: 99%

“…Owing to these facts, there is always the need to develop new numerical methods of solution and to improve on the existing ones. A good number of research work had ever since been carried out on this problem, each providing numerical means of solution, for instance in [8], Legendre polynomial is applied via comparison technique while David [5] deployed Finite element method (FEM) in solving the same problem. In a bid to enhance results many authors had also applied collocation method with different means of improving its solutions (see [4], [6], [9]).…”

Section: Introductionmentioning

confidence: 99%

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Comparative Study of the Effect of Different Collocation Points on Legendre-Collocation Methods of solving Second-order Boundary Value Problems

2013

IOSR-JM

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We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.

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Section: Legendre Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Comparative Study of the Effect of Different Collocation Points on Legendre-Collocation Methods of solving Second-order Boundary Value Problems

2013

IOSR-JM

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On the approximation of particular solution of nonhomogeneous linear differential equation with Legendre series

Patanarapeelert¹,

Patanarapeelert²

2016

j.appsci

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In this study, we propose an approximation method for particular solutions of the nonhomogeneous second-order differential equations by truncated Legendre series. Particulary, the govern problem is a linear differential equation with constant coefficients. The choice of series solutions depends upon the complementary solutions and the approximate nonhomogeneous terms. An upper bound for the approximation error is formulated. Some examples are presented to demonstrate the validity of the proposed method.Keywords: nonhomogeneous differential equations, particular solutions, Legendre polynomials, series, error bound IntroductionMany problems in natural science such as physics, engineering and mathematical modeling are governed by differential equations (Jung et al., 2014). Solving such equations will lead to understanding the behaviors of the systems. Although solutions are known to be exist, there is an only few problems that can be solved for analytic solution. Several attempts are devoted to numerical method or approximation techniques to obtain the high accuracy of approximations (Jung et al., 2014).Taylor series and orthogonal functions such as Chebyshev and Legendre polynomials are powerful tools for functions approximations in terms of polynomials (Gulsu et al., 2006; Wang and Xiang, 2012;Patanarapeelert and Varnasavang, 2013). As a by-product they can be used for approximating the solution of ordinary differential equations. Sezer and Gulsu (2010) proposed a numerical method based on the hybrid Legendre and Taylor polynomials for solving the high-order linear differential equations. Olagunju and Olaninejum (2012) formulated a trial solution for nonhomogeneous differential equations where Legendre polynomials are used as basis functions. Recently, Jung et al. (2014) proposed the method to solutions of second-order differential equations by using Tau method based on Legendre operational matrix.In this paper, we present a method for approximating the particular solutions of nonhomogenous linear second-order differential equations. Rather than approximating as a whole we focus on in part, approximating particular solution by which the complementary solution is prior known. In doing so, we transfer the original problem into an approximate one

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Legendre-coefficients Comparison Methods for the Numerical Solution of a Class of Ordinary Differential Equations

Cited by 2 publications

References 6 publications

Comparative Study of the Effect of Different Collocation Points on Legendre-Collocation Methods of solving Second-order Boundary Value Problems

Comparative Study of the Effect of Different Collocation Points on Legendre-Collocation Methods of solving Second-order Boundary Value Problems

On the approximation of particular solution of nonhomogeneous linear differential equation with Legendre series

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