engleski

A Collocation Method for Singularly Perturbed Two-Point Boundary Value Problems with Splines in Tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problem $ay""+by"+cy=f(cdot,y), y(a) = y_0, y(b) = y_1$. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems ($|a|=varepsilon ll 1$), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.

spline in tension; ordinary differential equation

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