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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.

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A third derivative method (TDM) with continuous coefficients is derived and used to obtain a main and additional methods, which are simultaneously applied to provide all approximations on the entire interval for initial and boundary value problems of the form y = f (x, y, y ). The convergence analysis of the method is discussed. An algorithm involving the TDMs is developed and equipped with an automatic error estimate based on the double mesh principle. Numerical experiments are performed to show efficiency and accuracy advantages.