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Zbl 0742.73005
Saldanha da Gama, Rogério Martins
Existence, uniqueness and construction of the solution of the energy transfer problem in a rigid and nonconvex black body. (English)
[J] Z. Angew. Math. Phys. 42, No.3, 334-347 (1991). ISSN 0044-2275; ISSN 1420-9039/e

The paper deals with the problem of steady heat conduction of a rigid, radiating black body. The radiation is by the well-known Stefan-Boltzmann law. The body is assumed to be nonconvex so that there exists a direct radiant energy exchange between the points on the boundary of the body. The heat conduction in the body is governed by a linear partial differential equation generated by Fourier's law. The boundary condition is represented by a nonlinear integro-differential operator acting on the temperature at the boundary whose kernel depends on the shape of the body and vanishes for convex bodies. It is shown by employing a variational principle that the solution for a nonconvex body is obtainable through a Cauchy sequence of solutions obtained by omitting radiant energy transfer. However, on evaluating the $i-1$st element of the sequence the ith element plays the part of a source term due to the nonconvexity of the body. By using some results of functional analysis it is proved that the sequence is convergent and there exists a unique limit function. Hence the author provides an iterative scheme for solving such problems.
[E.S.Suhubi (\D{I}stanbul)]

MSC 2000:: *74A15 Thermodynamics
74S30 Other numerical methods
74P10 Optimization of other properties
35Q72 Other PDE from mechanics
35K05 Heat equation

Keywords: radiative heat transfer; existence and uniqueness of physical solution; Stefan-Boltzmann law; linear partial differential equation; Fourier's law; nonlinear integro-differential operator; variational principle; Cauchy sequence of solutions

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Zentralblatt MATH Berlin [Germany]