Convergence of regularized spline approximants to solutions of initial and boundary value problems for ODE

Spline collocation has been known for some time as a very suitable numerical method in solving linear initial and boundary value problems for ODE (Voevudsky, 1989). In those cases, however, when the right-hand side of the equation, given on a discrete mesh, includes errors, the method for spline collocation becomes rather unstable with respect to the errors. ject of the present paper is to investigate the convergence of an approximate solution defined as a minimizer of Tikhonov-like regularizer. All results presented in this paper are valid both for initial value problems and, with slight modifications, for boundary value problems with zero boundary conditions; but we have chosen an initial value problem of the first order to simplify our considerations. shown that the spline approximant exists for any mesh and for any positive value of the regularization parameter and that it converges to the solution of the initial value problem if an appropriate regularization parameter is chosen. We have made use of some ideas of Ragozin (1983).