THIRD-DERIVATIVE BLOCK METHODS FOR DIRECT SOLUTION OF SECOND ORDER BOUNDARY VALUE PROBLEMS

Abstract

In this research, third derivative continuous block methods for direct solution of second order boundary value problems of ordinary differential equations (ODEs) with Neumann, Dirichlet and Robin boundary conditions are developed. The methods are considered for step numbers three, four and five respectively. This was achieved by constructing a continuousrepresentation of linear multistepschemesviainterpolationoftheapproximatesolutionandcollocationofderivativefunctionswithpowerseriesasbasisfunctions. Thediscreteschemeswereobtainedfromthecontinuous scheme as a by-product and applied in block form as simultaneous numerical integrators to solve boundary value problems. The resultant schemes are self-starting, do not need the development of separate predictors. The order and convergence analysis of the methods are studied. Numerical examples are considered and the results compared with those provided by existing works in the literature. The comparison shows that the derived methods Performed better in terms of accuracy than the existing methods