One-step hybrid block method with one generalized off-step points for direct solution of second order ordinary differential equations

Abdelrahim, Ra’ft; Omar, Zurni

doi:10.12988/ams.2016.6127

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…These numerical results are seen in Problem 1 where a linear IVP was considered. Problem 1 was solved by [13] and [14] where both studies developed hybrid block methods to solve this problem. Table 1 shows the newly developed method in this article having better accuracy, in terms of absolute error, over the results obtained by [13] and [14].…”

Section: Discussionmentioning

confidence: 99%

“…Problem 1 was solved by [13] and [14] where both studies developed hybrid block methods to solve this problem. Table 1 shows the newly developed method in this article having better accuracy, in terms of absolute error, over the results obtained by [13] and [14]. This improved accuracy is also seen in Figure 1 where the accuracy line representing the new method is closer to the value 0 than the other methods.…”

Section: Discussionmentioning

confidence: 99%

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Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Mansor¹,

Adeyeye²,

Omar³

2023

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The numerical of second order initial value problems (IVPs) has garnered a lot of attention in literature, with recent studies ensuring to develop new methods with better accuracy than previously existing approaches. This led to the introduction of hybrid block methods which is a class of block methods capable of directly solving second order IVPs without reduction to a system of first order IVPs. Its hybrid characteristic features the addition of off-step points in the derivation of this block method, which has shown remarkable improvement in the accuracy of the block method. This article proposes a new three-step hybrid block method with three generalized off-step points to find the direct solution of second order IVPs. To derive the method, a power series is adopted as an approximate solution and is interpolated at the initial point and one off-step point while its second derivative is collocated at all points in the interval to obtain the main continuous scheme. The analysis of the method shows that the developed method is of order 7, zero-stable, consistent, and hence convergent. The numerical results affirm that the new method performs better than the existing methods it is compared with, in terms of error accuracy when solving the same IVPs of second order ordinary differential equations.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Mansor¹,

Adeyeye²,

Omar³

2023

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“…Two years later, Omar and Abdelrahim developed a one‐step HBM with generalized three off‐steps using interpolation and collocation for solving second‐order initial‐value problems (IVPs). Similarly, Abdelrahim and Omar derived a one‐step HBM with generalized two off‐steps using interpolation and collocation for solving third‐order IVPs. The following year, Omar and Abdelrahim proposed a one‐step HBM including three off‐step points for the solution of general fourth‐order ODEs.…”

Section: Introductionmentioning

confidence: 99%

Heat transfer study of convective fin with temperature‐dependent internal heat generation by hybrid block method

Alkasassbeh

Omar

Mebarek‐Oudina

et al. 2019

Heat Trans. Asian Res.

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Purpose In this study, an implicit one‐step hybrid block method using two off‐step points involving the presence of a third derivative for solving second‐order boundary value problems are subjected to Dirichlet‐mixed conditions. Methodology To derive this method, the approximate power series solution is interpolated at { x n , x n + 2 5 } while its second and third derivatives are collocated at all points { x n , x n + 2 5 , x n + 3 5 , x n + 1 } on the integrated interval of approximation. Findings The new derived method not only performs better compared with the existing methods when solving the same problems but also obtains better properties of the numerical method. Afterward, the proposed method is applied to solve the problem of a convective fin with temperature‐dependent internal heat generation. The effects of various physical parameters on temperature distribution are also examined.

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Numerical Approximation for Direct Solution of Third-Order Ordinary Differential Equations Using New Class of Falkner-Type Block Method

S.¹

2023

Advanced Journal of Science, Technology and Engineering

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In this article, a new class of Falkner-type methods for direct solution of third order ordinary differential equation is developed. The approach of collocation and interpolation technique is adopted to derive the new Falkner-type methods, which is implemented in block mode to get approximation at grids points simultaneously. The resulting scheme is zero-stable, consistent and convergent with good region of absolute stability. The tabular and graphical presentations of the numerical results to the problems considered, demonstrate the effectiveness and good accuracy of the scheme in comparison with other methods

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One-step hybrid block method with one generalized off-step points for direct solution of second order ordinary differential equations

Cited by 4 publications

References 2 publications

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Numerical Solution of Linear and Nonlinear Second Order Initial Value Problems Using Three-Step Generalized Off-Step Hybrid Block Method

Heat transfer study of convective fin with temperature‐dependent internal heat generation by hybrid block method

Numerical Approximation for Direct Solution of Third-Order Ordinary Differential Equations Using New Class of Falkner-Type Block Method

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