The present study addresses forced convection heat transfer of an internal viscous fluid in a tube with fully developed laminar velocity and uniform entrance temperature. The internal viscous fluid exchanges heat with an external viscous fluid moving normally to the tube at a different temperature. Specifically, the description corresponds to a generalized Graetz problem with heat convection boundary condition. Contrary to the tradition in vogue, the standard method of separation of variables and the ensuing Sturm-Liouville theory are not employed for solving the generalized Graetz problem in the present study. Rather, the goal of the study is to implement an approximate finite difference methodology with an explicit scheme. The primary thermal quantity of interest in the study is the mean bulk temperature of the internal viscous fluid accounting for the entire range of modified Biot numbers (0 < Bi < infinity). Subsequently, the other thermal quantities of secondary interest in the study are the wall temperature and the total heat transfer. The agreement of the approximate numerical results with the counterpart exact analytical results is excellent for all values of Bi ranging from 0 to 100 in practice. The exact analytical results expressed in terms of the generalized Graetz series are considered as the baseline solutions in the heat convection literature.