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Abstract: This paper is concerned with the exact solutions for thermal diffusion in a straight fin with varying exponential shape when the thermal conductivity and heat transfer coefficients are temperature dependent. The conduction and heat transfer terms constitute the strong nonlinearity given by power laws. Explicit analytical solutions are derived for the relevant parameters in terms of special functions of advanced mathematics. The effects of parameters of physical interest such as the fin tip temperature, the fin efficiency and the fin base heat transfer rate can be better captured from the presented formulas. Moreover, with the help of the analysis here, the best shape of the straight fin is identified under a variety of operating conditions. Analysis further shows that the efficiency and base heat transfer rate of the exponential profiles are higher than those of the rectangular fin. [Copyright &y& Elsevier]