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.

In this paper, the conventional backward differentiation formulae methods for step numbers k = 3 and 4 were reformulated by shifting them one-step backward to produce two and three approximate solutions respectively, in a step when implemented in block form. The derivation of the continuous formulations of the reformulated methods were carried out through multistep collocation method by matrix inversion technique. The discrete schemes were deduced from their respective continuous formulations. The convergence analysis of the discrete schemes were discussed. The stability analysis of these schemes were ascertained and the P- and Q-stability were also investigated. When the discrete schemes were implemented in block form to solve some first order delay differential equations together with an accurate and efficient formula for the solution of the delay argument, it was observed that the results obtained from the schemes for step number k = 4 performed slightly better than the schemes for step number k = 3 when compared with the exact solutions. More so, on comparing these methods with some existing ones, it was observed that the methods derived performed better in terms of accuracy.

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This research work is aimed at constructing a class of explicit integrators with improved stability and accuracy by incorporating an off-gird interpolation point for the purpose of making them effcient for solving stiff initial value problems. Accordingly, continuous formulations of a class of hybrid explicit integrators are derived using multi-step collocation method through matrix inversion technique, for step numbers k = 2; 3; 4: The discrete schemes were deduced from their respective continuous formulations. The stability and convergence analysis were carried out and shown to be A(α)-stable and convergent respectively. The discrete schemes when implemented as block integrators to solve some non-linear problems, it was observed that the results obtained compete favorably with the MATLAB ode23 solver.

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