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Over a finite one dimensional specimen containing two phases of a pure substance, it has been shown that the liquid-solid interface motion exhibits parabolic behavior at small time intervals. We study the interface behavior over a finite domain with homogeneous Dirichlet boundary conditions for large time intervals, where the interface motion is not parabolic due to finite size effects. Given the physical nature of the boundary conditions, we are able to predict exactly the interface position at large time values. These predictions, that to the best of our knowledge, are not found in the literature, were confirmed by using the heat balance integral method of Goodman and a non-classical finite difference scheme. Using heat transport theory, it is shown as well, that the temperature profile within the specimen is exactly linear and independent of the initial profile in the asymptotic time limit. The physics of heat transport provides a powerful tool that is used to fine tune the numerical methods. We also found that in order to capture the physical behavior of the interface, it was necessary to develop a new non classical finite difference scheme that approaches asymptotically to the predicted interface position. We offer some numerical examples where the predicted effects are illustrated, and finally we test our predictions with the heat balance integral method and the non classical finite difference scheme by studying the liquid-solid phase transition in Aluminum.
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