Developing a single step hybrid block method with generalized three off-step points for the direct solution of second order ordinary differential equations

Omar, Zurni; Abdelrahim, Ra’ft

doi:10.12988/ijma.2015.57181

Cited by 5 publications

(2 citation statements)

References 3 publications

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“…However, the accuracy regarding error was not very efficient. 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.…”

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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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.

View full text Add to dashboard Cite

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“…In the literature, many researchers proposed the methods having the combination of hybrid and block methods. 6,7,[11][12][13]17,18,20,[31][32][33][34][35] Many authors proposed the hybrid methods with the arbitrary values to the off-step points (e.g., one can see , then the Ramos et al 35 first proposed the concept of finding the values to the off-step points, which improves the efficiency of the method in terms of accuracy.…”

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confidence: 99%

A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Singh

Garg²,

Singla

et al. 2021

Comp and Math Methods

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In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points x n+1∕2 and x n+1 .The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems. K E Y W O R D Sblock methods, embedded-type procedure, hybrid methods, ordinary differential equations INTRODUCTIONNumerous numerical integrators are available for solving first-order initial value ordinary differential systems numerically. Runge-Kutta and linear multistep methods are the usual classes of methods that are being used for solving these systems numerically. These days many built-in computer codes based on these classes of methods are available in software like Matlab, Mathematica, and so forth. To develop more accurate and efficient codes for solving the differential systems is an ongoing process of research. This article is concerned with the development of a class of optimized codes for solving the following system:whereTo proceed further, we assume that the system (1) has a unique continuously differentiable solution that we denote by Y(x). As a first step, the interval of interest [x 0 , x N ] is discretized as follows:x n = x 0 + nh, n = 0, 1, 2, 3, … , N; h = x n+1 − x n and an approximate numerical solution of the system (1) at x n is denoted as Y n ≈ Y(x n ).

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Investigation of a hyperbolic annular fin with temperature dependent thermal conductivity by two step third derivative block method (TSTDBM)

et al. 2020

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Developing a single step hybrid block method with generalized three off-step points for the direct solution of second order ordinary differential equations

Cited by 5 publications

References 3 publications

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

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

A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Investigation of a hyperbolic annular fin with temperature dependent thermal conductivity by two step third derivative block method (TSTDBM)

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