Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins

•Application of the two dimensional differential transform method.•New analytical solutions to nonlinear transient heat transfer in longitudinal rectangular and convex parabolic fins.•Study of the effects of thermal parameters on temperature distribution.•Comparison of the heat transfer in longitudinal rectangular against convex parabolic fins In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.