Bernstein induced one step hybrid scheme for general solution of second order initial value problems

Ojo Ezekiel Olukunle; M. Okoro Felix

doi:10.26637/MJM0802/0006

Abstract

In this paper, a Bernstein polynomial with collocation and interpolation techniques were used to develop one step hybrid scheme with one offgrid point for the direct solution of general second order ordinary differential equations. The basic properties of the derived scheme was investigated and found to be of order four(4), zero stable and convergent. The scheme obtained is used to solve some standard initial value problems. From the numerical results obtained, it was revealed that the proposed method performs better than some of the existing methods in the literature.

Keywords:

Bernstein polynomial, Collocation, nterpolation, nterpolationZero Stability, Consistency, Region of Absolute stability

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Ojo Ezekiel Olukunle Department of Mathematics, Ambrose Alli University, Ekpoma, Ekpoma, Nigeria.
M. Okoro Felix Department of Mathematics, Ambrose Alli University, Ekpoma, Ekpoma, Nigeria.

Pages: 350-355
Date Published: 01-04-2020
Vol. 8 No. 02 (2020): Malaya Journal of Matematik (MJM)

A. O. Adeniran, S. A. Odejide and B. S. Ogundare, One-Step Hybrid Numerical Scheme for Direct solution

of General Second Order Ordinary Differential Equations, International Journal of Applied Mathematics, 28(3)(2015), 197-212.

H. M. Ahmed, Solutions of 2nd-order linear differential equations subject to Dirichlet boundary conditions in a Bernstein polynomial basis, Journal of the Egyptian Mathematical Society, 22(2)(2014), 227-237.

S.B. Ahmad, Bernstein Polynomials Method and its Error Analysis for Solving Nonlinear Problems in the Calculus of Variations: Convergence Analysis via Residual Function, Published by Faculty of Sciences and Mathematics, University of Nis, Serbia, 32:4(2018), 1379-1393

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)(4)(2012), 65-75.

E. A. Areo and R. B. Adeniyi, A Self-Starting Linear Multistep Method for Direct Solution of Initial Value Problem of Second order Ordinary Differential Equations,International Journal of Pure and Applied Mathematics, 82(3)(2013), 345-364.

M.H.T. Alshbool, A.S. Bataineh, I. Hashim, and O.R. Isik, Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions, Journal of King Saud University Science, 29(2017), 1-18.

M.H.T. Alshbool, and I.Hashim, Multistage Bernstein polynomials for the solutions of the Fractional Order Stiff Systems, Journal of King Saud University Science, 2016, 280-285.

D. O. Awoyemi, A new sixth-order algorithm for general second order Ordinary Differential Equations, International Journal of Computer Mathematics, 2001, 117-124.

D. O. Awoyemi and S.J. Kayode, A maximal order collocation method for direct solution IVPs of general second ordinary differential equations, In Proceedings of the conference organized by the National Mathematical Center; Abuja, 2005.

A.D. Aysegul, and I. Nese, Bernstein Collocation Method for Solving Nonlinear Differential Equations, Numerical Computational Application, 18(3)(2013a), 293-300.

A.D. Aysegul, and I. Nese, Bernstein Collocation Method for Solving linear Differential Equations, Gazi University Journal of Science, 26(4)(2013b), 527-534.

M. I. Bhatti and P. Bracken, Solutions of differential equations in a Bernstein polynomial basis, Journal of Computational and Applied Mathematics, 205(1)(2017), $272-280$.

G. Dahlquist, On accuracy and unconditional stability of LMM for second order differential equations, BIT, 18(1978), 133-136.

S. Davaeifar, J. Rashidinia and M. Amirfakhrian, Bernstein Polynomial Approach for Solution of Higher-Order Mixed Linear Fredholm Integro-Differential Equation with Variable Coefficients, TWMS Journal of Pure Applied. Mathematics, 7(N.1)(2016), 46-62.

E. H.Doha, A. H. Bhrawy and M. A. Saker, On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations, Boundary Value Problems, Article ID 829543, 2011.

S. O. Fatunla, Block methods for second order IVPs, International Journal of Computer Mathematics, 41(1991), $55-63$.

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Englewood Cliffs, New Jersey, Prentice-Hall. 1971.

$[18]$ S. Hao , M. Yi, J. Huang, and Y. Pan, Bernstein Polynomials Method for a Class of Generalized Variable Order Fractional Differential Equations, IAENG International Journal of Applied Mathematics, 46(4)(2016), 46-50.

Joy, K.I. Bernstein Polynomials: [OnLine] Geometric Modeling Notes, Available: http://en.wikipedia.org/wiki/Bernstein Polynomials. 2002.

S. J. Kayode, and O. Adeyeye, A Two-Step Point Hybrid Method for General Second Order Ordinary Differential Equations, African Journal of Mathematics and Computer Science Research, 6(10)(2013), 191-196.

S.N. Khataybeh, I. Hashim and M. Alshbool, Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations, Journal of King Saud University-Science, 2018.

J. D. Lambert, Computational Methods in Ordinary Differential Equations, New York: John Wiley 1973.

J. D. Lambert, Numerical Methods for Ordinary Differential Systems, New York, John Wiley, 1991.

Y. Ordokhani and S.F. Davaei, Approximate solutions of differential equations by using the Bernstein polynomials, International Scholarly Research Network ISRN Applied Mathematics, 2011, 1-15.

M. J. Rupa and M. S. Islam, Numerical Solution of System of Second Order Boundary Value Problems Using Galerkin Method, GANIT J. Bangladesh Mathematics Society, 37(2017), 161-174.

A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Applied Mathematical Modelling, 38(4)(2014), 1365-1372.

Y. Salih and H.Gurler, Bernstein Collocation Method for Solving the First Order Nonlinear Differential Equation with the Mixed Non-Linear Conditions, Mathematical and Computational Applications, 20(3)(2015), 160-173.

O. A. Taiwo1 and M. O. Hassan, Approximation of Higher-order Singular Initial and Boundary Value Problems by Iterative Decomposition and Bernstein Polynomial Methods, British Journal of Mathematics & Computer Science, 9(6)(2015), 498-515.

Y. A. Yahaya and A. M. Badmus, A Class of Collocation Methods for General Second Order Ordinary Differential Equations, African Journal of Mathematics and Computer Science Research, 2(04)(2009), 69-72.

S. A. Yousefi, M. Behroozifar and M. Dehghan, The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass, Journal of Computational and Applied Mathematics, (2011), 5272-5283.