Continuous finite difference approximations for solving differential equations

Abstract

In recent papers continuous finite difference (FD) approximations have been developed for the solution of the initial value problem (ivp) for first order ordinary differential equations (odes). They provide dense output of accurate solutions and global error estimates for the ivp economically. In this paper we show that some continuous F D formulae can be used to provide a uniform treatment of both the ivp and the boundary value problem (bvp) without using the shooting method for the latter. Higher order accurate solutions can be obtained on the same meshes with constant spacing used by one-step method without using the iterated deferred correction technique. No additional conditions are required to ensure low order continuity and this leads to fewer necessary equations than those required by most of the popular methods for bvps. No quadratures are involved in this non-overlapping piecewise continuous polynomial technique. Some computed results are given to show the effectiveness of the proposed method and global error estimates.

An Eight Order Two-Step Taylor Series Algorithm for the Numerical Solutions of Initial Value Problems of Second Order Ordinary Differential Equations

“…Actually, considerable attention has been devoted to solving ordinary differential equation of higher order directly without reduction for instance: methods of linear multistep method (LMM) were considered by Lambert and Watson [8], Dormand and El-Mikkawy [9], El-Mikkawy and ElDesouky [10] and Awoyemi [11] [12] [13] [14]. Subsequently, LMM was independently proposed by Kayode [15], Onumanyi et al [16] and Adesanya et al [17] in the predictor-corrector mode, based on collocation method. These authors proposed LMM with continuous coefficients where they adopted Taylor series algorithm to supply the starting values.…”

Multiple Off-Grid Hybrid Block Simpson

“…For the discrete solution of (1) linear multi-step methods has being studied by [7], [8], and continuous solutions of (1) [9]and [12], [13]. One important advantage of the continuous over the discrete approach is the ability to provide discrete schemes for simultaneous integration.…”

“…By using [12], [14] approach, the derived schemes will be applied in block form in other to achieve its order and error constants; the region of absolute stability, and the results of absolute errors.…”

High Order Block Implicit Multi-Step (Hobim) Methods for the Solution of Stiff Ordinary Differential Equations

“…Though these yielded a very successful combination but have a draw back because of its requirements for starting values which could lead to growing numerical errors and corrupting further approximations Mayers and Suli [24]. To resolve the issue of starting value, Onimanyi etal [25], [26] proposed the block Linear Multistep methods based on the multi-step collocation approach of Lie and Norset [23] and the self-starting methods of Cash [6]. These methods were developed through the continuous formulation of the linear k-step methods which provided sufficient number of simultaneous discrete methods used as single integrators.…”