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.

No abstract

No abstract

SUMMARYCubic basis functions in one dimension for the solution of two-point boundary value problems are constructed based on the zeros of Chebyshev polynomials of the first kind. A general formula is derived for the construction of polynomial basis functions of degree r, where 1 Qr< co. A Galerkin finite element method using the constructed basis functions for the cases r = 1,2 and 3 is successfully applied to three different types of problem including a singular perturbation problem.

In this paper, we construct a family of high order self-starting one-block numerical methods for the solution of stiff initial value problems (IVP) in ordinary differential equations (ODE). The Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF) and Backward Differentiation Formulas (BDF) are used in the constructions. The E-transformation is applied to the triples and a family of self-starting methods are obtained. The family is for . The numerical implementation of the methods on some stiff initial value problems are reported to show the effectiveness of the methods. The computational rate of convergence tends to the theoretical order as h tends to zero.