# AN ORDER SEVEN BERNSTEIN INDUCED HYBRID TWO-STEP METHOD FOR DIRECT SOLUTION OF SECOND-ORDER INITIAL PROBLEMS

## Keywords:

Block method, Hybrid, Interpolation, Collocation, Bernstein polynomial## Abstract

In this article, a two step Bernstein Hybrid Multistep Method (BHMM) of order seven is developed for the direct solution of second order initial problems. To derive this method, the approximate function was interpolated and collocated at equidistant grid and offgrid points. The continuous scheme was evaluated at different off-step points to obtain multiple hybrid method and presented as a block method. The resulting methods are zero-stable, consistent and convergent. Some numerical examples were given to demonstrate the accuracy and efficiency of the proposed method and found to give better approximation than the existing methods.

**Views | Downloads:**54 / 9

## Downloads

## References

M. I. Bhatti, P. Bracken, Solutions of differential equations in a Bernstein polynomial basis, Journal of Computational and Applied Mathematics 205 (2007) 272-280.

S. Davaeifar, J. Rashidinia, M. Amirfakhrian, Bernstein Polynomial Approach for Solution of Higher- Order Mixed Linear Fredholm Integro-Differential-Difference Equations with Variable Coefficients, TWMS J. Pure Appl. Math. V.7, N.1, (2016), pp.46-62

D. O. Awoyemi, A New Sixth-order algorithm for general second order ordinary differential equation, International Journal of Computer Mathematics 77,(2001). 117-124

D. O. Awoyemi, & S. J. Kayode, A maximal order collocation method for direct solution of initial value problems of general second order ordinary differential equation, Proceeding of the Conference Organized by the National Mathematical Centre, Abuja, Nigeria (2005).

D. O. Awoyemi, A. O. Adesanya and S. N. Ogunyebi, Construction of self starting Numerov method for the solution of initial value problem of general second ordinary differential equation, Journ. Num. Math, 4(2) (2008) 267-278.

R. A. Bun, T.D. Vasils, A numerical method for solving differential equations of any Order, Comp.Math. Phys, 32(3) (1992) 317-330.

A. O. Adeniran, and I. O. Longe, One Step Trigonometrically-fitted Third Derivative Method with Oscillatory Solutions, Journal of Advances in Mathematics and Computer Science 25(6): (2017) 1-7.

N. M. Kamoh, A. A. Abada, and M. C. Soomiyol, A block procedure with continuous coefficients for the direct solution of general second order initial value problems of (ODEs) using shifted Legendre polynomials as basis function, International Journal of Multidisciplinary Research and Development, vol. 5 Issue 4,(2018) 236-241.

S. N. Jator, A sixth order linear multistep method for the direct solution of y′′ = f(x, y, y′), International Journal of Pure Applied Mathematics 40(1), (2007) 457-472.

J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley and Sons, London New York Sydney Toronto 1973.

M. Alkasassbeh, and Z. Omar, Generalized two-hybrid one-step implicit third derivatives block method for the direct solution of second order ordinary differential equations, International Journal of Pure and Applied Mathematics 112(3) (2017) 497-517

B. T. Olabode and A. L. Momoh, Chebyshev Hybrid Multistep Method for Directly Solving Second- Order Initial and Boundary Value Problems, Journal of the Nigerian Mathematical Society, Vol. 39, Issue 1, (2020) 97-115.

A. O. Adeniran, and I. O. Longe, Solving Directly Second Order Initial Value Problems with Lucas Polynomial, Journal of Advances in Mathematics and Computer Science, 32(4): (2019) 1-7.

L. A. Ukpebor, A 4-Point Block Method for Solving Second Order Initial Value Problems in Ordinary Differential Equations, American Journal of Computational and Applied Mathematics, 9(3) (2019) 51- 56

E. O. Ojo and F. M. Okoro, A Block Scheme with Bernstein Single Step Method for Direct Solution Second Order Differential, International Journal of Research and Innovation in Applied Science (IJRIAS) Vol.IV, Issue XII, Dec. (2019) 99-103.

E. O. Ojo and F. M. Okoro, Bernstein induced one step hybrid scheme for general solution of second order initial value problems, Malaya Journal of Matematik, Vol. 8, No. 2, (2020) 350-355.

S. O. Fatunla, Block methods for second order IVPs, Int. J. Comput. Maths.41 No 9 (1991) 55-63.

P. Henrici, Discrete Variable Methods for ODEs, John Wiley, New York, USA (1962).

Ojo, Adoghe and Okoro J. of NAMP J. D. Lambert, and A. Watson Symmetric multistep method for periodic initial value problem, Journal of

the Institute of Mathematics and its Applications, 18 (1976), 189-202.

A.O. Adesanya, M.R. Odekunle, Continuous Block Hybrid-Predictor-Corrector method for the solution of y′′ = f(x, y, y′), International Journal of Mathematics and Soft Computing Vol.2, No.2, (2012) 35 - 42.

T. A. Anake, D. O. Awoyemi and A. O. Adesanya, One step implicit hybrid block method for Direct solution of General Second Order Ordinary Differential Equations, International Journal of Applied Mathematics 42 2 No 4 (2012) 65-75.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2023 The Journals of the Nigerian Association of Mathematical Physics

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## How to Cite

*The Journals of the Nigerian Association of Mathematical Physics*,

*64*, 65–74. https://nampjournals.org.ng/index.php/home/article/view/82