Solutions of the extended Graetz problem

In this paper laminar forced convective heat transfer problems inside ducts, with axial conduction, subjected to the three main types of boundary conditions are solved exactly. The general method of solution involves a change of the dependent variable leading to a square integrable function in the real line. A complete basis for the vector space of these functions is used to generate an infinite expansion for the solution. The form of solutions is presented for the flows inside a circular pipe, the annular space between pipes, and between parallel plates.