Open AccessDevelopment and Implementation of Block Unification Multi-step Methods for the Solution of Second Order Ordinary Differential Equations
Author(s) -
Usman Mohammed,
Oyelami Oyewole,
Mikhail Semenov,
A. I. Ma’ali
Publication year - 2019
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1366/1/012088
Subject(s) - mathematics , integrator , ordinary differential equation , collocation (remote sensing) , block (permutation group theory) , boundary value problem , ode , convergence (economics) , collocation method , function (biology) , orthogonal collocation , mathematical optimization , differential equation , mathematical analysis , computer science , computer network , geometry , bandwidth (computing) , machine learning , evolutionary biology , economic growth , economics , biology
In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found in the literature.