Abstract
It is the aim of this paper to give some insight how fixed point principles work to develop results in pure analytical as well as in numerical respect on the field of free boundary problems for partial differential equations. In the beginning a series of examples is presented where free boundaries become involved in all three classical types of partial differential equations elliptic, hyperbolic and parabolic. Later on equations of parabolic type only are studied in detail. It is shown how Schauder's fixed point theorem can be applied to prove existence in melting problems as well as in a model describing the mixture of different fluids. Numerical experiments confirm that these methods can also be useful to obtain practical results.
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Hoffmann, K.H. (1981). Fixpunktprinzipien und Freie Randwertaufgaben. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090681
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DOI: https://doi.org/10.1007/BFb0090681
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