Formation of hybrid block method of higher step-sizes, through the continuous multi-step collocation

Ibijola, E. A.; Skwame, Y.

doi:10.5251/ajsir.2011.2.2.161.173

Cited by 8 publications

(15 citation statements)

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“…Note that, which is also the expression for the mass of the substance at any time t. Applying our new half step numerical scheme (8) to solve the Growth model (22) gives results as shown in Table 3 [16,17] (Table 3). …”

Section: Problem 3 (Decay Model)mentioning

confidence: 99%

“…We also assume that   N t is the population size at time ( Table 2). t The theoretical solution of (18) Applying our new half step numerical scheme (8) to solve the Growth model (17) gives results as shown in Table 2 [16].…”

Section: Problem 2 (Growth Model)mentioning

confidence: 99%

“…This block method has the properties of Runge-kutta method for being self-starting and does not require development of separate predictors or starting values. Among these authors are [7][8][9][10][11][12]. Block method was found to be cost effective and gave better approximation.…”

Section: Introductionmentioning

confidence: 99%

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Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations

James¹,

Adesanya²,

Sunday³

et al. 2013

AJCM

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In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent and zero stable hence convergent. The new method was tested on real life problems namely: SIR model, Growth model and Mixture Model. The results were found to compete favorably with the existing methods in terms of accuracy and error bound.

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Section: Problem 3 (Decay Model)mentioning

confidence: 99%

Section: Problem 2 (Growth Model)mentioning

confidence: 99%

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Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations

James¹,

Adesanya²,

Sunday³

et al. 2013

AJCM

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“…Where y satisfies a given set of initial condition (Ibijola, et al, 2011), and we assume that the function f also satisfies the Lipschitz condition which guarantees existence, uniqueness and continuous differentiable solution, [6]. For the discrete solution of (1) linear multi-step methods has being studied by [7], [8], and continuous solutions of (1) [9]and [12], [13].…”

Section: Introductionmentioning

confidence: 99%

“…Block methods for solving ordinary differential equations have initially been proposed by [10] who advanced their use only as a means of obtaining starting values for predictor-corrector algorithms. Several authors [3], [4], [5], [11], [13,[15], [16], [ 17] among others, have modified it to be more efficient as a computational procedure for the integration of IVPs throughout the range of integration rather than just as a starting method for method for multistep methods [1].…”

Section: Introductionmentioning

confidence: 99%

Multiple Off-Grid Hybrid Block Simpson

2016

IJSR

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This paper is concerned with the construction of two-step hybrid block Simpson's methods with two and three off-grid points for the solutions of stiff systems of ordinary differential equations (ODEs). This is achieved by transforming a k-step multi-step method into continuous form and evaluating at various grid points to obtain the discrete schemes. The discrete schemes are applied each as a block for simultaneous integration. Each block matrix equation is A-stable and of order [5,5,5,6] T and [6,6,6,6,6] T respectively. These orders are achieved by the aid of Maple13 software program. The performance of the methods is demonstrated on some numerical experiments. The results revealed that the hybrid block Simpson's method with two-off grid points was more efficient than that hybrid block method with three off-grid points on mildly stiff problems.

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Block multistep methods based on rational approximants

Ying¹,

Omar²,

Mansor³

2014

AIP Conference Proceedings

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Variable step direct block multistep method for general second order ODEs AIP Conf.A rational approximation for the parabolic equation method A parabolic wave equation based on a rational-cubic approximation J. Acoust. Soc. Am. 87, 619 (1990); 10.1121/1.398930 N−variable rational approximants and method of moments Abstract. In this study, the concept of block multistep methods based on rational approximants is introduced for the numerical solution of first order initial value problems. These numerical methods are also called rational block multistep methods. The main reason to consider block multistep methods in rational setting, is to improve the numerical accuracy and absolute stability property of existing block multistep methods that are based on polynomial approximants. For this pilot study, a 2-point explicit rational block multistep method is developed. Local truncation error and stability analysis for this new method are included as well. Numerical experimentations and results using some test problems are presented. Numerical results are satisfying in terms of numerical accuracy. Finally, future issues on the developments of rational block multistep methods are discussed.

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Formation of hybrid block method of higher step-sizes, through the continuous multi-step collocation

Cited by 8 publications

References 0 publications

Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations

Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations

Multiple Off-Grid Hybrid Block Simpson

Block multistep methods based on rational approximants

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