The Heat Equation Solution Near the Interface Between Two Media

Physical  phenomena  that arise near the boundaries  of media with different characteristics, for example,  changes in temperature at  the  water-air interface,  require  the  creation  of models for their  adequate description. Therefore,  when setting  model problems  one should  take  into  account the fact  that the  environment parameters undergo  changes  at  the  interface.   In particular, experimentally obtained temperature curves at the water-air interface have a kink, that is, the derivative  of the temperature  distribution function  suffers a discontinuity at the interface.  A function  with this feature  can be a solution to the problem for the heat equation  with a discontinuous thermal diffusivity and discontinuous function  describing  heat  sources.  The  coefficient of thermal diffusivity  in the  water-air transition layer is small, so a small parameter appears  in the  equation  prior  to the  spatial  derivative, which makes the equation  singularly  perturbed.  The  solution  of the  boundary value  problem  for such an equation  can have the  form of a contrast structure, that is, a function  whose domain  contains  a subdomain, where the  function  has a large gradient. This  region is called an internal  transition layer.  The  existence  of a solution with the internal  transition layer of such a problem requires justification that can be carried out with the use of an asymptotic analysis.  In the present paper,  such an analytic  investigation was carried out,  and this  made it possible to prove the  existence  of a solution  and also to construct its asymptotic approximation.

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