Greenʹs functions, temperature and heat flux in the rectangle

Steady heat conduction in the rectangle is treated with the method of Greenʹs functions. Single-sum series for the Greenʹs functions are reported in terms of exponentials which have better numerical properties than hyperbolic functions. Series expressions for temperature and heat flux caused by spatially uniform effects are presented. The numerical convergence of these series is improved, in some cases by a factor of 1000, by replacing slowly converging portions of the series with fully summed forms. This work is motivated by high-accuracy verification of finite-difference and finite-element codes.