Science Domain

 Journal

 One-step Hybrid Block Method For Directly Solving Fifth-order Initial Value Problems Of Ordinary Differential Equations

 M. K. Duromola , A. L. Momoh And J. M. Adeleke

 2022

An effective one-step hybrid block for getting the approximate solution of a fifth-order IVP with applications to problems in the sciences and engineering is constructed in this study. The mathematical formulation of the method is based on the principle of interpolation and collocation of the trial solution and its derivatives at the chosen equidistant grid and off-grid points. The basic properties of the derived method are examined, and it has an order greater than one, zero stable, consistent, and hence convergent. The derived method is applied to solve five different linear and nonlinear fifth-order initial value problems. Comparison of the absolute errors obtained using the derived method with a few existing ones in the literature supports its good performance. 

 Inderscience

 Journal

 Insight Into 2-step Continuous Block Method For Solving Mixture Model And SIR Model

 M.K. Duromola; A.L. Momoh; M.A. Rufai; I.L. Animasaun

 2021

Understanding of the solutions of first-order ordinary differential equations (ODEs), mixture model and SIR model in order to develop deep insight and exploration are major problems before the experts, biologists, scientists, and mathematicians. In all these problems, the governing equations are either single first-order or coupled ODEs kind of initial value problem (IVP). In this paper, a polynomial function q ( x ) that passes through the points ( x n , y n ), ( x n +1 , y n +1), . . . , ( x n +2 , y n +2) was adopted as the basis function that leads to third derivatives continuous 2-step block method suitable to solve first order initial value problems of ODEs. Upon using the newly proposed scheme to solve linear ODEs (i.e., mixture theory) and nonlinear ODE (i.e., SIR model), it is worth concluding that the algorithm is not only efficient but minimises error. 

 Scientific Research

 Journal

 Hybrid Numerical Method With Block Extension For Direct Solution Of Third Order Ordinary Differential Equations

 M. K. Duromola, A. L. Momoh

 2019

This paper focuses on the development of a hybrid method with block extension for direct solution of initial value problems (IVPs) of general third-order ordinary differential equations. Power series was used as the basis function for the solution of the IVP. An approximate solution from the basis function was interpolated at some selected off-grid points while the third derivative of the approximate solution was collocated at all grid and off-grid points to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was implemented with five test problems including the Genesio equation to confirm its accuracy and usability. The rate of convergence (ROC) reveals that the method is consistent with the theoretical order of the proposed method. Comparison of the results with some existing methods shows the superiority of the accuracy of the method. 

 American Journal

 Journal

 A ZERO -STABLE HYBRID LINEAR MULTISTEP METHOD FOR THE NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

 Bolarinwa Bolaji; Monday Duromola

 2017

In this paper, a new Zero – stable continuous hybrid linear multistep method is proposed for the numerical solution of initial value problems of first order ordinary differential equations. The procedure for derivation of the numerical method entails obtaining the main method and additional schemes from the same continuous scheme derived via interpolation and collocation procedure. The implementation of the method is by applying the main scheme together with the additional schemes in a block form as simultaneous integrators over non – overlapping intervals. The method was found to be consistent, zero – stable, convergent and accurate. 

 American Journal

 Journal

 A ZERO -STABLE HYBRID LINEAR MULTISTEP METHOD FOR THE NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

 Bolarinwa Bolaji; Monday Duromola

 2017

In this paper, a new Zero – stable continuous hybrid linear multistep method is proposed for the numerical solution of initial value problems of first order ordinary differential equations. The procedure for derivation of the numerical method entails obtaining the main method and additional schemes from the same continuous scheme derived via interpolation and collocation procedure. The implementation of the method is by applying the main scheme together with the additional schemes in a block form as simultaneous integrators over non – overlapping intervals. The method was found to be consistent, zero – stable, convergent and accurate. 

 IOSJOURNALR

 Journal

 Derivation Of One-Sixth Hybrid Block Method For Solving General First Order Ordinary Differential Equations

 Rufai M. A.1 , Duromola M.K.2 , And Ganiyu A.A.

 2016

This paper focuses on the derivation of one-sixth hybrid block method for the general solution of first order initial values problems of ordinary differential equations. The new proposed method was derived by using the approach of collocation and interpolation of Chebyshev polynomials, approximate solution at some selected points to get a continuous linear multistep method, which was evaluated at some off-grid points to generate hybrid linear multistep methods. Basic properties of the proposed method wasexamined and the method found to be zero-stable, consistent and convergent. The efficiency of the method was tested on some numerical examples and in particular, on well-known SIR Model, Prothero-Robinson oscillatory problem and highly stiff oscillatory problem. On comparison, the new proposed method performed favourably when compare with the existing method proposed by other researchers in the area of Numerical Analysis. 

 AENSI Journals

 Journal

 Modified Block Method For The Direct Solution Of Initial Value Problems Of Fourth Order Ordinary Differential Equations.

 R.A. Ademiluyi; M.K. Duromola; B. Bolarinwa

 2014

In this article, we present a new block method for the direct solution of initial value problems of fourth order ordinary differential equations. The approach of collocation approximation is adopted in the derivation of the main scheme with continuous coefficients, from where additional schemes were developed. The implementation strategy is by combining the main scheme and the additional schemes as simultaneous integrator to initial value problem of fourth order ordinary differential equations. Properties analysis of the block showed that it is consistent, convergent, zero stable and absolutely stable. Our method was tested with numerical examples solved using existing method and was found to give better results. 

 SCIENCEDOMAIN International

 Journal

 Direct Solution Of Initial Value Problems Of Fourth Order Ordinary Differential Equations Using Modified Implicit Hybrid Block Method

 S.J. Kayode; M.K. Duromola; B. Bolarinwa

 2014

Our focus in this article is the derivation; analysis and implementation of a new modified implicit hybrid block method for the direct solution of initial value problems of fourth order ordinary differential equations. In the derivation of the method, we adopted the approach of collocation approximation to obtain the main scheme with continuous coefficients. From the main scheme, additional schemes were developed. The implementation strategy of the new method is by combining the main scheme and the additional schemes as simultaneous integrator to initial value problem of fourth order ordinary differential equations. As required of any numerical method, the properties analysis of the block was done and the result showed that it is consistent, convergent, zero stable and absolutely stable. We then test our method with numerical examples solved using existing method and were found to give better results. 

 Scientific Research

 Journal

 An Accurate Five Off-Step Points Implicit Block Method For Direct Solution Of Fourth Order Differential Equations

 M. K. Duromola,

 2006

In this article, my focus is the derivation, analysis and implementation of a new modified one-step implicit hybrid block method with five off-step points. The derived method is to solve directly initial value problems of fourth order ordinary differential equations. The approach for the derivation of the method is to interpolate the approximate power series solution to the problem and to collocate its fourth derivative at the grid and off-grid points to generate systems of linear equations for the determination of the unknown parameters. The derived method is tested for consistency, zero stability, convergence and absolute stability. Accuracy and usability of the method are determined with some test problems and the results obtained are found to be better in accuracy than some existing methods.