DevelopmentandImplementationofBlockUnificationMulti-stepMethodsfortheSolutionofSecond OrderOrdinaryDifferentialEquations

Abstract

In this paper, linear multi-step hybrid block methods with three-, four- and fivestep 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 constructedasbasisfunctiontodevelopthehybridblockmethodsadoptingcollocationtechniquewithanon-negative weightfunction. Theschemeisappliedassimultaneousintegratortosecondorderinitialvalueandboundaryvalue 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.