Download Case Studies in Industrial Mathematics by Heinz W. Engl, Ewald Lindner (auth.), a. Univ. Prof. PDF

By Heinz W. Engl, Ewald Lindner (auth.), a. Univ. Prof. Dipl.-Ing., Dr. Heinz W. Engl, o. Univ. Prof. Dr. Hansjörg Wacker, Dipl.-Ing. Dr. Walter Zulehner (eds.)

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3) f g dS ClG 0, b. the solution is unique in E; c. the solution depends continuously on the data g. 15) resp. 16). Therefore, the problem of calculating the potentials ~i and xi' i = 1,2,3, is well posed: A unique solution exists, which continuously depends on the data. The hydrodynamic coefficients can be expressed by the form a(u,v) (up to some physical constants): G~. 1. 2) is the unique minimizer of the form f(u) 1 2 a(u,u) - l(u). Proof: see [ 3] . 37 More can be said for bodies of revolution.

V dS l(v) E aG tV E H 1 (G) J V dx O} G where H1 (G) denotes the Sobolev space of all functions on G with square-integrable (weak) first derivatives. 2) is called a weak solution of the von Neumann boundary value problem. e. g is square- integrable on ClG). Then a. 3) f g dS ClG 0, b. the solution is unique in E; c. the solution depends continuously on the data g. 15) resp. 16). Therefore, the problem of calculating the potentials ~i and xi' i = 1,2,3, is well posed: A unique solution exists, which continuously depends on the data.

Because of the fact that solidification takes place over a relatively large temperature interval, it is natural to use the enthalpy for modelling our problem. 1) H(u)= u f uo ps{T)dT+pL(u), where u 0 is some reference temperature. If ~ 1 ,u 2 J is the temperature range within which solidification takes place, then L is a monotone function with for u ( 2. 2) < u1 for u 2 u 2 . gradu), 3t where k=k(u) denotes the thermal conductivity at temperature u, t denotes time and div and grad are taken with respect to the space variables.

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