A theoretical study of convective heat transfer is presented for a laminar flow subjected to an axial variation in the external heat transfer coefficient (or dimensionless Biot number). Since conventional techniques fail for a variable boundary condition parameter, a variable eigenfunction approach is developed. An analysis is carried out for a periodic heat transfer coefficient, which serves as a model for heat transfer from a duct fitted with an array of evenly spaced fins. Three solution methods for the variable eigenfunction technique are examined: an Nth order approximation method, an iterative method and a stepwise periodic method. The stepwise periodic method provides the most convenient and accurate solution for a stepwise periodic Biot number. Graphical results match exactly to ones obtained by Charmchi and Sparrow from a finite-difference scheme. A connected region technique is also developed to provide limited exact results to test the validity of the three solution methods. The study of a finned duct by a stepwise periodic Biot number is carried out via a parametric study, an average (constant) Biot number approximation and an assumed velocity profile analysis. Results for the parametric study show that external finning yields substantial heat transfer enhancement over an unfinned duct, especially when the Biot number of the unfinned regions is low. A decrease in the interfin spacing causes increased enhancement. Variations of the period of the Biot number causes relatively small changes in enhancement as long as the ratio of finned to unfinned surface remains unchanged. An average (constant) Biot number approximation for a specified finned tube is compared to the stepwise periodic Biot number solution. The results show that the constant Biot number approximation provides accurate results. Finally, the results for the influence of the assumed velocity profile demonstrate that a constant velocity flow provides increased heat transfer and more effective enhancement by external finning than a laminar fully developed flow, especially at high Biot numbers. This study provides insight into heat transfer enhancement due to finning and also develops a solution methodology for problems involving variable boundary condition parameters. / M.S.