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We discuss a parameter estimation method which can be used to estimate functional parameters in delay differential equations and moving boundary problems. In either problem, we approximate the original model equation (which is infinite dimensional) with a system of ordinary differential equations that can be solved numerically in an efficient way. The approximation scheme is based on time-dependent spline elements. For the delay equation with time-varying delay, we present convergence results which indicate that the estimates obtained using the approximating system converge in some sense to a best-fit parameter for the original system. We present numerical test examples in which we estimate time-varying and state-dependent delays, and in which we estimate a time-varying diffusion coefficient in a one phase, one dimensional Stefan problem.