Two point hermite approximations for the solution of linear initial value and boundary value problems

Abstract

Application of an idea originally due to Ch. Hermite allows the derivation of an approximate formula for expressing the integral ∫ x i x i −1 y( x) dx as a linear combination of y( x i −1), y( x i ), and their derivatives y ( v)( x i −1) up to order v = α and y ( v) ( x i ) up to order v = β. In addition to this integro-differential form a purely differential form of the 2-point Hermite approximation will be derived. Both types will be denoted by H αβ -approximation. It will be shown that the well-known Obreschkoff-formulas contain no new elements compared to the much older H αβ -method. The H αβ -approximation will be applied to the solution of systems of ordinary differential equations of the type y'( x) = M( x) y( x) + q( x), and both initial value and boundary value problems will be treated. Function values at intermediate points x ϵ ( x i−1 , x i ) are obtained by the use of an interpolation formula given in this paper. An advantage of the H αβ -method is the fact that high orders of approximation (α, β) allow an increase in step size h i . This will be demonstrated by the results of several test calculations.