The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

Skwame, Y.; Sabo, J.; Mathew, Mobi

doi:10.9734/ajpas/2019/v5i330136

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…Consequently, the solution of ( 1) is simultaneously generate on the entire interval of integration by using the main method (10) for n = 0, 1, ..., N 2 to obtain 3N − 2 equations and the additional method (11) and (12) which are employed to complete the set of equations needed to simultaneously solve 3N by 3N system of equations for solving (1) directly.…”

Section: Implementation Tacticsmentioning

confidence: 99%

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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: Implementation Tacticsmentioning

confidence: 99%

“…On the other hand, the shooting method suffers inaccuracy and instability while the finite different method is very demanding and does not give a satisfactory results. Other numerical methods for solving ordinary differential equations also exist in literature [5][6][7][8][9][10][11][12][13][14][15].…”

Section: Introductionmentioning

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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“…Skwame et al developed an equidistant one-step block hybrid method for treating second-order ordinary diferential equations. Te developed one-step block hybrid method was applied to a real-world physics and engineering problem, namely, the cooling of a body in reference [4]. Next, the authors of reference [5] proposed a new trigonometrically two-step hybrid method for solving second-order ODEs with one or two frequencies.…”

Section: Introductionmentioning

confidence: 99%

A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems

Rufai

Shokri

Omole

2023

Journal of Mathematics

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This paper describes a third-derivative hybrid multistep technique (TDHMT) for solving second-order initial-value problems (IVPs) with oscillatory and periodic problems in ordinary differential equations (ODEs), the coefficients of which are independent of the frequency omega and step size h . This research is significant because it has numerous applications to real-life phenomena such as chaotic dynamical systems, almost periodic problems, and duffing equations. The current method is derived from the collocation of a derivative function at the equidistant grid and off-grid points. The TDHMT obtained is a continuous scheme for obtaining simultaneous approximations to the solution and its derivative at each point in the x 0 , x N interval integration. The presence of high derivatives increases the order of the method, which increases the accuracy method’s order and the stability property, as discussed in detail. Finally, the proposed method is compared to existing methods in the literature on some oscillatory and periodic test problems to demonstrate the technique’s effectiveness and productivity.

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“…Equation (1) has a wide range of applications because many problems that are encountered in sciences, real-life, control theory, and engineering are modelled into Differential Equations. This is why the numerical solution of ( 1) is of great interest to researchers such as [2].…”

Section: Introductionmentioning

confidence: 99%

Development of an Order (k+3) Block-Hybrid Linear Multistep Method for the Direct Solution of General Second Order Initial Value Problems

MURITALA¹,

Kolawole²,

OYEDEJİ³

et al. 2022

UJEES

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Block hybrid linear multistep method was proposed to overcome the Dahl Quist order barrier for linear multistep methods. This research aims to answer questions relating to the convergence, accuracy, and effectiveness of the block hybrid method when utilized to obtain the solution of Initial Value Problems (IVPs). In this research, an order (k+3) block hybrid method applicable to obtain the direct solution of IVP’s of ordinary differential equations (ODEs) is presented. Collocation and interpolation of power series at finely selected grid points were used to improve the method’s consistency, convergence, accuracy and zero stability. Linear problems were solved to show the accuracy and efficiency of the proposed method, and the error obtained from the comparison of exact and approximate results shows that the proposed method is effective in solving the class of problem.

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The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

Cited by 5 publications

References 11 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

A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems

Development of an Order (k+3) Block-Hybrid Linear Multistep Method for the Direct Solution of General Second Order Initial Value Problems

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