Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems

Adeyeye, Oluwaseun; Omar, Zurni

doi:10.1016/j.asej.2017.11.011

Cited by 11 publications

(13 citation statements)

References 14 publications

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“…In engineering and natural sciences, implicit RK-type methods are still widely used, especially for solving stiff systems [34,35]. As researchers are familiar with Butcher tableaus, the existing program codes can be enhanced in accuracy according to the proposed approach by forming an altered set of equations which is a rather small modification that retains the number of unknowns.…”

Section: Discussionmentioning

confidence: 99%

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Urevc

Halilovic̆

2021

Mathematics

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In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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Section: Discussionmentioning

confidence: 99%

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Urevc

Halilovic̆

2021

Mathematics

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“…Hence, the block algorithm is convergent since it is both consistent and zero-stable. Error in [13] Error in [12] block method…”

Section: Convergencementioning

confidence: 99%

“…Error in [12] PC method 0.103125 -0.10715827398597420496 -0.10715827398597420499 3.0000e-20 [12,13] Figure 1. Graphical error of our method with [12,13] when solving system 1 2.7553e-06 Source [6] Figure 3. Graphical error of our method with [6] when solving system 3…”

Section: Convergencementioning

confidence: 99%

Numerical Application of Higher Order Linear Block Method for Solving Some Fourth Order Initial Value Problems

Raymond¹,

Kyagya²,

John³

et al. 2023

Afr. Sci. Rep.

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The numerical application of higher order linear block method for the direct solution of fourth order initial value problems was proposed using the linear block algorithm, where the methods applied in block form. The method is zero-stabile, consistent and convergent when analyzing the properties of the method. The mathematical example solved using the method is effective, suitable, and acceptable for solving fourth order initial value problems. The method is also compared with existing work when solving similar systems of differential equation and obviously, the method performs better than those in literature and textual shown.

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“…[8], proposed a numerical solution for fourth-order initial value problems using lucas polynomial with application in ship dynamics. The numerical approximations of (1) have been considered in literature by authors such as [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

On the Simulation of Higher Order Linear Block Algorithm for Modelling Fourth Order Initial Value Problems

Sabo

Skwame²,

Donald³

2022

ARJOM

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The introduction of new linear block method for the direct simulation of fourth order IVPs has been developed in this article. The reason for adopting direct simulation of fourth order initial value problems is to address some setbacks in reduction method. When developing the method, we adopted the linear block approach through a one step method. We have validated the accuracy of the method on some fourth order initial value problems without reduction process, and the results are better than the conventional method. The numerical experiments were given and the results obtained were found to be better in accuracy than the existing methods in literature.

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Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems

Cited by 11 publications

References 14 publications

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Numerical Application of Higher Order Linear Block Method for Solving Some Fourth Order Initial Value Problems

On the Simulation of Higher Order Linear Block Algorithm for Modelling Fourth Order Initial Value Problems

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