Continuous Block Method for the Solution of Second Order Initial Value Problems of Ordinary Differential Equation

James, Adewale; Adesanya, A. O.; Sunday, J.

doi:10.12732/ijpam.v83i3.3

Cited by 21 publications

(18 citation statements)

References 10 publications

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“…The name of this class of models derives from the fact that they involves coupled equations relating the number of susceptible people ( ) S t , number of people infected ( ) I t and the number of people who have recovered ( ) R t . This is a good and simple model for many infectious diseases including measles, mumps and rubella [13][14][15]. The SIR Trapezoidal Multistep Method of Second Kind for Stiff ODEs model is described by the three coupled equations.…”

Section: Implementation Strategiesmentioning

confidence: 99%

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

Awari¹

2017

SJAMS

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A Modified ThreeStep Block Hybrid Extended Trapezoidal Multistep Method of Second Kind (BHETR 2 s) with two off-grid points, one at interpolation and another at collocation point yielding uniform order six (6,6,6,6,6) T for the Numerical Integration of initial value problems of stiff Ordinary Differential Equations was developed. The main method and additional equations were obtained from the same continuous formulation through interpolation and collocation procedures. The stability properties of the method was discussed and from the stability region obtained, the method is suitable for the solution Stiff Ordinary Differential Equations. Three numerical examples were considered to illustrate the efficiency and accuracy.

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Section: Implementation Strategiesmentioning

confidence: 99%

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

Awari¹

2017

SJAMS

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“…Numerical solution of equations of this form is a very keen area of interest for researchers in literature, ranging from first order IVPs [6,11] to higher order IVPs [7,13]. The focus of this article however is on higher order IVPs (specifically, second order IVPs).…”

Section: Introductionmentioning

confidence: 99%

“…This is seen recent works of [6,9,14]. However, this article introduces another approach different from what is existing in literature on how hybrid block methods can also be developed.…”

Section: Introductionmentioning

confidence: 99%

4 -step 5-point hybrid block method for the direct solution of second order initial value problems

Adeyeye¹

2017

J. Math. Computer Sci.

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Block methods for the numerical solution of ordinary differential equations (ODEs) are quite prominent in recent literature and second order initial value problems (IVPs) which falls in the family of ODEs is also a well explored area for the application of block methods. The introduction of hybrid block method methods for the solution of second order IVPs has gained good grounds in literature as the presence of off-grid points in the block method has increased the accuracy of the hybrid block methods. However, recent studies still continue to introduce new block methods that will perform more favourably than previously existing when compared in terms of error. Hence, a hybrid block method of order six is presented in this article to compete with previously existing methods of the same order and higher order. The methodology adopted in this article presents a new approach for developing the hybrid block method which is simple to implement and less computationally tiresome. The numerical results show this new 4-step 5-point hybrid block method performing better than previously existing methods.

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“…Conventionally, the approach for obtaining a numerical approximation to equations of the form (1) above (also classified as higher order ordinary differential equations) usually involves a rigorous reduction of the higher order differential equations to a system of first order differential equations (James et al, 2013;Kayode and Adeyeye, 2011;Jator, 2007). Block multistep methods have evolved with time for the direct solution of second order initial value problems (Abdelrahim and Omar, 2016;Jator, 2007;Adesanya et al, 2012) and the concept of introducing evaluation at offgrid points birthing the introduction of hybrid block methods for numerical approximation of (1) have also been investigated into.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Block Method for Direct Numerical Approximation of Second Order Initial Value Problems Using Taylor Series Expansions

Adeyeye¹,

Omar²

2017

American Journal of Applied Sciences

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In this article, a hybrid block method is utilized for the numerical approximation of second order Initial Value Problems (IVPs). The rigor of reduction to a system of first order initial value problems is bypassed as the hybrid block method directly solves the second order IVPs. Likewise, the methodology utilized also avoids the cumbersome steps involved in the widely adopted interpolation approach for developing hybrid block methods as a simple and easy to implement algorithm using the knowledge from the conventional Taylor series expansions with less cumbersome steps is introduced. To further justify the usability of this hybrid block method, the basic properties which will infer convergence when adopted to solve differential equations are investigated. The hybrid block method validates its superiority over existing methods as seen in the improved accuracy when solving the considered numerical examples.

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Continuous Block Method for the Solution of Second Order Initial Value Problems of Ordinary Differential Equation

Cited by 21 publications

References 10 publications

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs

4 -step 5-point hybrid block method for the direct solution of second order initial value problems

Hybrid Block Method for Direct Numerical Approximation of Second Order Initial Value Problems Using Taylor Series Expansions

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