Fin equation represents an important model describing the heat distribution along the fins surface. The current research investigated different cases of variable thermal conductivity invoking the effect of fin parameter, N. Group theoretical method and hidden symmetry technique to analyze and obtain new closed form solutions to understand the heat distribution along the fins surface. The shape of the fins is the responsible of the variation of the fin parameter. Three cases were investigated for the thermal conductivity, constant, power law, and exponential cases. Some new solutions were obtained including Kummer and imaginary error functions. The analysis showed the high effect the fin parameter and the power. Increasing the fin parameter increases the heat distribution along the fin surface and decreases with the passage of time.