Multistep Collocation Block Method for Direct Solution of Second Order Ordinary Differential Equations

Kuboye, J. O.; Omar, Zurni

doi:10.3844/ajassp.2015.663.668

Cited by 3 publications

(6 citation statements)

References 5 publications

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“…Table 2 and Figure 2 also show the newly developed method performing better in the comparison of absolute error over [15] and [16]. Specifically, the method in [15] displayed inconsistency as seen in the graph in Figure 2. A nonlinear problem was considered in Problem 3 and the results from the newly developed method were compared with the block method by [15] and the collocation method by [17].…”

Section: Discussionmentioning

confidence: 82%

“…This implies that the new method had better accuracy at all points. For Problem 2, the solution of another linear IVP was compared to [15] and [16] where multistep block methods were developed by authors. Table 2 and Figure 2 also show the newly developed method performing better in the comparison of absolute error over [15] and [16].…”

Section: Discussionmentioning

confidence: 99%

“…For Problem 2, the solution of another linear IVP was compared to [15] and [16] where multistep block methods were developed by authors. Table 2 and Figure 2 also show the newly developed method performing better in the comparison of absolute error over [15] and [16]. Specifically, the method in [15] displayed inconsistency as seen in the graph in Figure 2.…”

Section: Discussionmentioning

confidence: 99%

“…Specifically, the method in [15] displayed inconsistency as seen in the graph in Figure 2. A nonlinear problem was considered in Problem 3 and the results from the newly developed method were compared with the block method by [15] and the collocation method by [17]. Although the methods by [15] and [17] competed closely to the new method in this article, there is still a clear improvement in accuracy by the new method in this article.…”

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: 82%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Concerning the 4-step block method, different formulations may be found in [21][22][23]. In [24][25][26][27] different formulations for the 5-step block method have appeared. For the 6-step block method one can see the references [28][29][30][31].…”

Section: K-mentioning

confidence: 99%

Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

Ramos

Jator

Modebei³

2020

Mathematics

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There are dozens of block methods in literature intended for solving second order initial-value problems. This article aimed at the analysis of the efficiency of k-step block methods for directly solving general second-order initial-value problems. Each of these methods consists of a set of 2k multi-step formulas (although we will see that this number can be reduced to k+1 in case of a special equation) that provides approximate solutions at k grid points at once. The usual way to obtain these formulas is by using collocation and interpolation at different points, which are not all necessarily in the mesh (it may also be considered intra-step or off-step points). An important issue is that for each k, all of them are essentially the same method, although they can adopt different formulations. Nevertheless, the performance of those formulations is not the same. The analysis of the methods presented give some clues as how to select the most appropriate ones in terms of computational efficiency. The numerical experiments show that using the proposed formulations, the computing time can be reduced to less than half.

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One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

Akinnukawe,

Okunuga

2024

J. Nig. Soc. Phys. Sci.

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In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

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Multistep Collocation Block Method for Direct Solution of Second Order Ordinary Differential Equations

Cited by 3 publications

References 5 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

Efficient k-Step Linear Block Methods to Solve Second Order Initial Value Problems Directly

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

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