A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

Osa, Adoghe Lawrence; Omole, Ezekiel Olaoluwa

doi:10.11648/j.acm.20190803.11

Cited by 15 publications

(14 citation statements)

References 9 publications

(9 reference statements)

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“…In comparison to Kuboye and Omar [8], Awoyemi et al [15], Adoghe and Omole [17] and Adeyeye and Omar [18], the hybrid block technique has shown improved accuracy with fewer steps. Furthermore, when compared to past higher-order techniques, the unique hybrid block strategy outperforms them.…”

Section: Discussionmentioning

confidence: 99%

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Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

Obarhua

2023

ARJOM

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This research work considers derivation of three step four point optimized hybrid block method for solving general third order differential equations (odes) without reduction to systems of lower order odes. A combination of power series and exponential function is used as an approximate solution to the general third order ode problems. Continuous linear multistep method is developed by interpolating the basis function at both grid and off-grid points and collocating the differential function at only grid points. The unknown parameters in the system of linear equations arising from the collocation and interpolation functions were determined and the values substituted in the approximate solution to the problem. The required continuous method is obtained after necessary simplification. The derived method is tested and found to be consistent, symmetric and of low error constant. The results obtained showed a better performance than the existing methods in literature under review.

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Section: Discussionmentioning

confidence: 99%

“…The absolute errors ec yy  obtained with the method for problem 1 is compared with that of [8], 10-step and [15], 5-step and order of accuracy 8 and 9 respectively. The absolute errors ec yy  obtained with the method for problem 2 is compared with that of [16] and [17] 2step and 8-step respectively.…”

Section: Region Of Absolute Stabilitymentioning

confidence: 99%

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

Obarhua

2023

ARJOM

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“…Theorem 3.1 [6] The compulsory and adequate terminologies for the optimized scheme to be convergent are that they must be consistent and zero-stable. Hence, the optimized scheme derived is convergent since all conditions are satisfied.…”

Section: Consistencymentioning

confidence: 99%

“…However, this process can only compute the numerical solution at one point at a time and time constraint [2,3]. Therefore, some scholars who newly applied the direct method to overcome the difficulties in reduction process in literature include [4,5,6]. Some researchers have proposed some methods in literature for solving (1), viz.…”

Section: Introductionmentioning

confidence: 99%

An Optimized Half-Step Scheme Third derivative Methods for Testing Higher Order Initial Value Problems

et al. 2023

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An optimized half-step third derivative block scheme on testing third order initial value problems is presented in this article. This scheme suggests some certain points of evaluation which properly optimizes the truncation errors at point of formulas, the conditions that guarantee the properties of the method was considered and satisfied. However the develop scheme is used to test some third order optimized problems and the mathematical outcomes achieved confirms better calculation than the previous method we related with.

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“…Many Authors such as Ogunware and Omole [3]; Adoghe and Omole [4]; Ukpebor, et al [5]; Olanegan, et al [6]; Adeyeye and Kayode [7]; Jator and Lee [8]; Hussain, et al [9] have devoted lots of attention to the development of various methods for solving (1) directly without reducing it to system of first order.…”

Section: Introductionmentioning

confidence: 99%

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

Ukpebor

Omole

Adoghe

2020

International Journal of Mathematical Research

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From the time immemorial, researchers have been beaming their search lights round the numerical solution of ordinary differential equation of initial value problems. This was as a result of its large applications in the area of Sciences, Engineering, Medicine, Control System, Electrical Electronics Engineering, Modeled Equations of Higher order, Thin flow, Fluid Mechanics just to mention few. There are a lot of differential equations which do not have theoretical solution; hence the use of numerical solution is very imperative. This paper presents the derivation, analysis and implementation of a class of new numerical schemes using Lucas polynomial as the approximate solution for direct solution of fourth order ODEs. The new schemes will bridge the gaps of the conventional methods such as reduction of order, Runge-kutta's and Euler's methods which has been reported to have a lot of setbacks. The schemes are chosen at the integration interval of seven-step being a perfection interval. The even grid-points are interpolated while the odd grid-points are collocated. The discrete scheme, additional schemes and derivatives are combined together in block mode for the solution of fourth order problems including special, linear as well as application problems from Ship Dynamics. The analysis of the schemes shows that the schemes are Reliable, P-stable and Efficient. The basic properties of the schemes were examined. Numerical results were presented to demonstrate the accuracy, the convergence rate and the speed advantage of the schemes. The schemes perform better in terms of accuracy when compared with other methods in the literature. Contribution/Originality: The study uses Lucas polynomial for the derivation of a new class of numerical schemes. The schemes were implemented in block mode for approximating fourth order ODEs directly without reduction. It solves variety of problems including problem in Electrical Engineering. The schemes performs excellently better than other schemes in the literatures.

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A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

Cited by 15 publications

References 9 publications

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

Three-step Four-point Optimized Hybrid Block Method for Direct Solution of General Third Order Differential Equations

An Optimized Half-Step Scheme Third derivative Methods for Testing Higher Order Initial Value Problems

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

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