Heat transfer in the thermal entrance region of a pipe or a parallel plate channel has been analysed for laminar and turbulent internal flow taking into account axial heat conduction effects in the fluid. The present paper shows an analytical solution for the problem of a piecewise uniform wall heat flux. The obtained exact analytical solutions for the extended Graetz problem are as simple and efficient to compute as the related solutions of the parabolic problem. The obtained results show the effect of axial heat conduction in the fluid for a semi-infinite as well as for a finite length of the heated sections.