An Efficient Seven-Step Block Method for Numerical Solution of SIR and Growth Model

In this article, a new implicit continuous block method is developed using the interpolation and collocation techniques via Power series as the basis function. A constant step length within a seven-step interval of integration was adopted. The selected grid points were evaluated to get a continuous linear multistep method. The evaluation of the continuous method at the non-interpolation points produces the discrete schemes which form the block. The basic properties of the block method were investigated and found to be consistent, zero stable and hence convergent. The new method was tested on real life problems namely: SIR and Growth model. The results were found to compare favourably with the existing methods in terms of accuracy and efficiency. Keywords: Block method, Growth Model, implicit, power series and SIR model.

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Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

Journal of the Nigerian Society of Physical Sciences ◽

10.46481/jnsps.2021.140 ◽

2021 ◽

pp. 26-37

Author(s):

Olumide O. Olaiya ◽

Rasaq A. Azeez ◽

Mark I. Modebei

Keyword(s):

Power Series ◽

Method Of Lines ◽

Second Order ◽

Basis Functions ◽

Direct Solution ◽

Block Method ◽

Partial Differential ◽

Integration Schemes ◽

The Method Of Lines ◽

Grid Points

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derivedschemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature.

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Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

FUOYE Journal of Engineering and Technology ◽

10.46792/fuoyejet.v5i2.537 ◽

2020 ◽

Vol 5 (2) ◽

Author(s):

Emmanuel A Areo ◽

Nosimot O Adeyanju ◽

Sunday J Kayode

Keyword(s):

Initial Value Problems ◽

Hybrid Methods ◽

Multistep Method ◽

Second Order ◽

Block Method ◽

Initial Value ◽

Order Error ◽

Block Methods ◽

Grid Points ◽

Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order

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Four-Step One Hybrid Block Methods for Solution of Fourth Derivative Ordinary Differential Equations

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2021/v36i330343 ◽

2021 ◽

pp. 1-10

Author(s):

Raymond, Dominic ◽

Skwame, Yusuf ◽

Adiku, Lydia

Keyword(s):

Approximate Solution ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Multistep Method ◽

Linear Multistep Method ◽

Block Method ◽

Step Method ◽

Block Methods ◽

Fourth Derivative ◽

Continuous Block

We consider developing a four-step one offgrid block hybrid method for the solution of fourth derivative Ordinary Differential Equations. Method of interpolation and collocation of power series approximate solution was used as the basis function to generate the continuous hybrid linear multistep method, which was then evaluated at non-interpolating points to give a continuous block method. The discrete block method was recovered when the continuous block was evaluated at all step points. The basic properties of the methods were investigated and said to be converge. The developed four-step method is applied to solve fourth derivative problems of ordinary differential equations from the numerical results obtained; it is observed that the developed method gives better approximation than the existing method compared with.

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A Seven-Step Block Multistep Method for the Solution of First Order Stiff Differential Equations

Momona Ethiopian Journal of Science ◽

10.4314/mejs.v12i1.5 ◽

2020 ◽

Vol 12 (1) ◽

pp. 72-82

Author(s):

Solomon Gebregiorgis ◽

Hailu Muleta

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Power Series ◽

Differential Equations ◽

Multistep Method ◽

Stiff Differential Equations ◽

Block Method ◽

Initial Value ◽

Numerical Examples ◽

First Order

In this paper, a seven-step block method for the solution of first order initial value problem in ordinary differential equations based on collocation of the differential equation and interpolation of the approximate solution using power series have been formed. The method is found to be consistent and zero-stable which guarantees convergence. Finally, numerical examples are presented to illustrate the accuracy and effectiveness of the method. Keywords: Power series, Collocation, Interpolation, Block method, Stiff.

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DERIVATION AND IMPLEMENTATION OF A THREE-STEP HYBRID BLOCK ALGORITHM (TSHBA) FOR DIRECT SOLUTION OF LINEAR AND NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

FUDMA Journal of Sciences ◽

10.33003/fjs-2020-0404-505 ◽

2021 ◽

Vol 4 (4) ◽

pp. 477-483

Author(s):

O. E. Abolarin ◽

B. G. Ogunware ◽

A. F. Adebisi ◽

S. O. Ayinde

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Conditions ◽

Multistep Method ◽

Second Order ◽

Direct Solution ◽

Block Method ◽

Order Error ◽

Block Algorithm ◽

Grid Points

The development and application of an implicit hybrid block method for the direct solution of second order ordinary differential equations with given initial conditions is shown in this research. The derivation of the three-step scheme was done through collocation and interpolation of power series approximation to give a continuous linear multistep method. The evaluation of the continuous method at the grid and off grid points formed the discrete block method. The basic properties of the method such as order, error constant, zero stability, consistency and convergence were properly examined. The new block method produced more accurate results when compared with similar works carried out by existing authors on the solution of linear and non-linear second order ordinary differential equations

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Approximating non linear higher order ODEs by a three point block algorithm

PLoS ONE ◽

10.1371/journal.pone.0246904 ◽

2021 ◽

Vol 16 (2) ◽

pp. e0246904

Author(s):

Ahmad Fadly Nurullah Rasedee ◽

Mohammad Hasan Abdul Sathar ◽

Khairil Iskandar Othman ◽

Siti Raihana Hamzah ◽

Norizarina Ishak

Keyword(s):

Numerical Methods ◽

Differential Equations ◽

Numerical Algorithm ◽

Computational Cost ◽

Real Life ◽

Higher Order ◽

Multistep Method ◽

Stability And Convergence ◽

Block Algorithm ◽

Life Problems

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

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Solution for nonlinear Duffing oscillator using variable order variable stepsize block method

MATEMATIKA ◽

10.11113/matematika.v33.n2.1015 ◽

2017 ◽

Vol 33 (2) ◽

pp. 165 ◽

Cited By ~ 2

Author(s):

Ahmad Fadly Nurullah Rasedee ◽

Mohamad Hasan Abdul Sathar ◽

Norizarina Ishak ◽

Nur Shuhada Kamarudin ◽

Muhamad Azrin Nazri ◽

...

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Duffing Oscillator ◽

Computational Cost ◽

Real Life ◽

Multistep Method ◽

Variable Order ◽

Block Method ◽

Step Size ◽

Variable Stepsize

Real life phenomena found in various fields such as engineering, physics, biology and communication theory can be modeled as nonlinear higher order ordinary differential equations, particularly the Duffing oscillator. Analytical solutions for these differential equations can be time consuming whereas, conventional numerical solutions may lack accuracy. This research propose a block multistep method integrated with a variable order step size (VOS) algorithm for solving these Duffing oscillators directly. The proposed VOS Block method provides an alternative numerical solution by reducing computational cost (time) but without loss of accuracy. Numerical simulations are compared with known exact solutions for proof of accuracy and against current numerical methods for proof of efficiency (steps taken).

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The Double Step Hybrid Linear Multistep Method for Solving Second Order Initial Value Problems

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v15i230145 ◽

2019 ◽

pp. 1-11

Author(s):

Y. Skwame ◽

J. Z. Donald ◽

T. Y. Kyagya ◽

J. Sabo

Keyword(s):

Approximate Solution ◽

Initial Value Problems ◽

Multistep Method ◽

Second Order ◽

System Of Nonlinear Equations ◽

Linear Multistep Method ◽

Block Method ◽

Initial Value ◽

Double Step ◽

Grid Points

In this paper, we develop the double step hybrid linear multistep method for solving second order initial value problems via interpolation and collocation method of power series approximate solution to give a system of nonlinear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic numerical properties of the hybrid block method was established and found to be zero-stable, consistent and convergent. The efficiency of the new method was conformed on some initial value problems and found to give better approximation than the existing methods.

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