In this paper we present a novel procedure for the determination of temperature in electric conductors. A Helmholtz-to-Poisson estimate is proved, that justifies to restrict the temperature dependence of the electrical resistivity to the conductor boundary. Hence we obtain a nonlinear potential problem for the relevant boundary temperatures, where the temperature dependence of the heat transfer coefficient is fully regarded. Using boundary integral operators, we represent the unknowns as the fixed point of a contraction. Finally a benchmark example is given in the rotationally symmetric case.
Dvorsky, K., Gwinner, J., & Liess, H.-D. (2011). A fixed point approach to stationary heat transfer in electric cables. Mathematical Modelling and Analysis, 16(2), 286-303. https://doi.org/10.3846/13926292.2011.580436
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