At the micro- and nanoscale, the standard continuity boundary conditions at fluid–solid interfaces of classical transport phenomena do not apply and must be replaced by boundary conditions that allow discontinuities. In this study, the classical thermal laminar boundary layer equations are studied using Lie symmetries with the no-slip boundary condition for tangential velocity and continuous temperature boundary conditions replaced by non-linear slip–creep–jump boundary conditions. These boundary conditions contain an arbitrary index parameter, denoted by n > 0, which appears in the coefficients of the coupled ordinary differential equations to be solved. As an independent check on the numerical procedure, the case of a boundary layer formed in a convergent channel with a sink, which corresponds to n = 1/2, is solved analytically for various values of the Prandtl number and zero Brinkham number. Other values of n for n > 1/2 which correspond to the thermal boundary layer formed in the flow past a wedge are solved numerically for various values of the Prandtl and Brinkham number and constant coefficients appearing in the non-linear slip–creep–jump boundary conditions. It is found that for 1/2 < n < 2, solutions may be found for all values of the constant coefficients, while for n ≥ 2 the constant coefficient for the creep term must be set to zero.

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