Improving convergence of summations in heat conduction

This paper addresses exact, transient heat-conduction solutions in two-dimensional rectangles heated at a boundary. The standard method of separation of variables (SOV) solution has two parts, steady-state (or quasi-steady) and complementary transient. The steady-state component frequently converges slowly at the heated surface, which is usually the one of greatest interest. New procedures are given to construct a steady solution in the form of a single summation, one having eigenvalues in the homogeneous direction (yielding the same result as the standard SOV solution) and the other having eigenvalues in the non-homogeneous direction (called the non-standard solution). The non-standard solutions have much better convergence behavior at and near the heated boundary than the standard forms. Examples are given.