On polynomials construction

In this work, an orthogonal polynomial with weight function w(x) =x2 + x + 1 in the interval [-1, 1] was constructed and used as the basis function to develop block methods, using collocation and interpolation approach. An efficient class of continuous and discrete numerical integration schemes of implicit hybrid form for third-order problems were developed and successfully implemented. Three different problems were solved with these schemes and they performed favourably. The investigation, using the appropriate existing theorems, shows that the methods are consistent, zero-stable and hence, convergent.

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“…where α j (x) and β j (x) are continuous coefficients. In (6) the parameters, α j (x) and β j (x), are obtained as :…”

Section: Numerical Block Algorithms For Third-order Problemsmentioning

confidence: 99%

“…Many researchers have used this block method to integrate IVPs, among them are [2], [3], [4], and [5]. Some researchers also constructed orthogonal polynomials using different weight functions; see [6]. These constructed polynomials were employed as basis function for the derivation of numerical schemes.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations

Joseph

Olagunju

Adeyefa

et al. 2022

J. Nig. Soc. Phys. Sci.

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“…The numerical solution of ODEs by collocation methods has been studied (see [1], [2], [3], [4], [5], [6], [7]). In the recent years, the solutions of (1) for cases m = 1, 2 & 3 have been extremely discussed (see [8], [9], [10], [11], [12]). The approach of reducing the higher order of (1) to a system of first order and the consequent setback has also been discussed in [13].…”

Section: Introductionmentioning

confidence: 99%

“…Solving(9) and(10) and substituting the outcomes into (8), we have) 11 ( .... .......... .......... .......…”

mentioning

confidence: 99%

A new class of orthogonal polynomials as trial function for the derivation of numerical integrators

Joseph¹

2022

GSC Adv. Res. Rev.

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This paper presents a set of newly constructed polynomials valid in interval [-1, 1] with respect to weight function w(x) = x + 1. For applicability sake, the polynomials shall be employed as trial function to develop a fast, efficient and reliable block algorithm for the numerical solution of ordinary differential equations with application to second order initial value problems. Collocation and interpolation techniques were adopted for the formulation of self-starting continuous hybrid schemes. Findings from the analysis of the basic properties of the method using appropriate existing theorems show that the developed schemes are consistent, zero-stable and hence convergent. On implementation, the superiority of the scheme over the existing method is established numerically. Further investigation of the properties of these polynomials is ongoing as we hope to discuss this in the future paper.

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On polynomials construction

Cited by 2 publications

References 1 publication

Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations

Hybrid Block Methods with Constructed Orthogonal Basis for Solution of Third-Order Ordinary Differential Equations

A new class of orthogonal polynomials as trial function for the derivation of numerical integrators

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