This book owes its origins to a scheme developed at the Mathematical Institute in Oxford to encourage the application of mathematics to problems occurring in the real world, in particular in industry. The realisation that perhaps a quarter of the differential equations problems arising in this scheme were to be solved in a region or domain which was not known in advance came as a surprise to everyone involved in view of the general lack of coverage at student level of these "free and moving boundary problems." In the past two decades there has, however, been a burgeoning of the mathematical literature on the subject and our principal aim in writing this book is to distil and explain what seem to be the most useful parts of the theory from the practical point of view.
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