AbstractIn this paper, we consider a mathematical model, which has a unique mechanism of heat transfer in the stretching/shrinking straight fin with an exponential profile. The thermal conductivity, internal heat generation, and heat transfer coefficient are considered temperature‐dependent. Heat is exposed to the surroundings by convection and radiation. The governing differential equation and boundary conditions are presented in a dimensionless form. In our study, we considered variable surface emissivity, that is, a constant, and the linear function of a temperature. The convective heat transfer parameter is considered a power‐low type. The novelty of this work is the application of temperature‐dependent surface emissivity, and the problem is solved by the Legendre wavelet collocation method. A comparative analysis of the present results in the context of previous findings is presented in the form of a table for validation and found exactly the same. The impacts of distinct variables are presented in the form of figures and discussed in detail. The present analysis is focused on real‐world applications and offers valuable insights for improving the design of fins.