Integral transforms in the two-dimensional non-linear formulation of longitudinal fins with variable profile

The integral transform method is employed for the hybrid numerical-analytical solution of two-dimensional, steady-state heat conduction within extended surfaces of variable longitudinal profile and temperature dependent thermal conductivity. Numerical results are then obtainable with automatic accuracy, allowing for the establishment of benchmark results and for the validation of approximate solutions. Convergence rates are illustrated for longitudinal fins with trapezoidal and parabolic profiles, and for different values of the governing parameters, Blot number and aspect ratio. In addition, the classical one-dimensional approximate solutions are critically examined for these typical non-straight profiles, and the applicability limits are investigated.