Numerical Treatment of General Third Order Ordinary Differential Equations Using Taylor Series as Predictor

Ogunware, Bamikole Gbenga; Awoyemi, D. O.; Adoghe, Lawrence Osa; Olanegan, O. O.; Omole, Ezekiel Olaoluwa

doi:10.9734/psij/2018/22219

Cited by 5 publications

(3 citation statements)

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“…Hence, F = 0, 0, 1 The block method of the form ( 10) is said to be zero stable if as ρ (F) = 0, then F j ≤ 1, j = 0, 1, ... for those roots with F j = 1, the multiplicity does not exceed 1 [21][22][23][24][25][26]. Also the block method (10) is consistent since p>1.…”

Section: Zero-stability Of the Hbmmentioning

confidence: 99%

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

Familua

Omole

Ukpebor

2022

J. Nig. Soc. Phys. Sci.

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In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multiderivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves.

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Section: Zero-stability Of the Hbmmentioning

confidence: 99%

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

Familua

Omole

Ukpebor

2022

J. Nig. Soc. Phys. Sci.

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“…Therefore, it is required to develop a numerical method that eliminates the utilization of predictor with improved accuracy, efficiency and that is selfstarting. Quite a number of researchers afterwards created block methods that addressed some of the mishaps of the predictor-corrector methods, [15][16][17]. Individuals worked on block methods using various approximation solution, which proved that the block methods are more reliable, efficient and give better accuracy.…”

Section: Original Research Articlementioning

confidence: 99%

The Simulation of One-Step Algorithms for Treating Higher Order Initial Value Problems

Sabo

Ayinde

Ishaq

et al. 2021

ARJOM

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The simulation of one-step methods using interpolation and collocation for the treatment of higher order initial value problems is proposed in this paper. The new approach is derived using interpolation and collocation as a basic function through power series polynomial, where the basic properties are also analyzed. The derived method is used to treat some highly stiff linear problems. The new approach compute clearly showed that the method is reliable, efficient and gives faster convergence when compared with those in literature.

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“…However, only a limited number of analytical methods are available for solving (1), hence the resort to numerical approximation methods. The well-known conventional method for solving (1) is to reduce it to a system of first order differential equations, Fatunla 1988) The reduction of such problems of type (1) to systems of first-order equations, leads to serious computational burden as well as wastage in computer time It has been reported that direct method for solving (1) is more efficient than the method of reduction to system of first order ordinary differential equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Implicit linear multistep methods which have better stability properties than explicit methods have been developed for the solution of (1) above [3-6, 10, 13, 17], among others proposed multi-derivative linear multistep method were implemented in predictor-corrector mode.…”

Section: Introductionmentioning

confidence: 99%

A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

Osa¹,

Omole²

2019

ACM

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In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were investigated and found to be zero stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing method.

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Numerical Treatment of General Third Order Ordinary Differential Equations Using Taylor Series as Predictor

Cited by 5 publications

References 0 publications

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

The Simulation of One-Step Algorithms for Treating Higher Order Initial Value Problems

A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

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