Verification solution for partial heating of rectangular solids

One method for verification of numerical solutions in heat transfer uses exact solutions for multi-dimensional parallelepipeds. The usual solution of these problems utilizes the method of separation of variables for both the steady and transient parts of the solution; however, this method for the steady-state part often produces solutions that converge slowly at the boundaries of greatest interest. Steady-state solutions having low convergence are illustrated for a rectangle with zero prescribed temperatures except one surface having a step change of temperature. Two forms of the steady-state solution are available and even then difficulties may be encountered. Another method called time-partitioning is used and is shown to have several superior features in the solution of this problem.