Hybrid One Step Block Method for the Solution of Fourth Order Initial Value Problems of Ordinary Differential Equations

Adesanya, A. O.; Fotta, A.U.; Abdulkadri, B.

doi:10.12732/ijpam.v104i2.1

Cited by 6 publications

(18 citation statements)

References 7 publications

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“…The numerical results were compared with the existing methods. The new method performed better than the methods in [10,[12][13][14], which employed 5 and 6 steps (refer to Tables 1 and 2). This implies that better accuracy can be achieved when step number k is increased.…”

Section: Resultsmentioning

confidence: 99%

“…Table 1 Comparison of new method with [13] and [14] for solving problem 1. Table 2 Comparison of new method with [10] and [12] for solving problem 2. Table 4 Comparison of new method with [7] for solving problem 4.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…(1) can be solved by reducing it to its equivalent first order system as mentioned in [1][2][3][4][5][6][7][8][9]. However, this approach suffers some setbacks, such as evaluation of too many functions and heavy computation (see [10][11][12]). …”

Section: Introductionmentioning

confidence: 99%

“…(1) have been examined by several researchers [10,[12][13][14]. They developed linear multistep methods using interpolation and collocation whereby the use of the approximated power series as a basis function was considered.…”

Section: Introductionmentioning

confidence: 99%

“…Kayode [12] developed an efficient zero-stable numerical method with step number k = 4 and 5 for fourth order initial value problems (IVPs), which was implemented in predictor-corrector mode. Furthermore, a five-step block method for solving fourth order ODEs directly is presented in [10]. In addition, in [13] and [14] six-step block methods are developed for solving fourth order ODEs using a multistep collocation approach whereby collocation points are selected at some grid points.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations

Omar

Kuboye

2016

j.math.fund.sci.

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Abstract:A new numerical method for solving fourth order ordinary differential equations directly is proposed in this paper. Interpolation and collocation were employed in developing this method using seven steps. The use of the approximated power series as an interpolation equation was adopted in deriving the method. The basic properties of the new method such as zero-stability, consistency, convergence and order are established. The numerical results indicate that the new method gives better accuracy than the existing methods when it is applied to fourth ordinary differential equations.

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Section: Resultsmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations

Omar

Kuboye

2016

j.math.fund.sci.

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Block Nystrom type method and its block extension for fourth order initial and boundary value problems

Adeyefa¹,

Akintunde²,

Ndu³

et al. 2018

ijma

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The derivation of Block Nyström type Method (BNTM) which is not normally used as numerical integrator of boundary value problems 184 E.O. Adeyefa et al. (BVPs) is considered and directly applied to solve both initial value problems (IVPs) and BVPs in ordinary differential equations (ODEs). Collocation technique is adopted in the derivation of the BNTM which is applied as simultaneous integrator to fourth order ODEs. The BNTM possesses the desirable feature of being self-starting as the implementation is in block form. The paper concludes by solving Numerical examples which establish the effectiveness and accuracy of the method. The superiority of BNTM is established by the numerical values presented.

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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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Hybrid One Step Block Method for the Solution of Fourth Order Initial Value Problems of Ordinary Differential Equations

Cited by 6 publications

References 7 publications

New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations

New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations

Block Nystrom type method and its block extension for fourth order initial and boundary 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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