Abstract: Based on the principle of the inverse solution to the undetermined boundary in complex heat transfer problem, a numerically stepwise approximation method was formed to calculate the erosion boundary values of the lining for blast furnace hearth via the procedure of regular solution-approval-boundary modification-secondary regular solution/approval. The method was formed in case there are only few even no temperature measuring points in blast furnace hearth and under the approved conditions that the measured temperature and heat flux in cooling stave are taken as erosion boundary values according to the axisymmetric steady-state heat transfer equation. On the principle of maximum risk, the method judges the non-uniqueness and ill-posed condition of the erosion profiles of axial lining sections so as to diagnose the blast furnace hearth and bottom. The diagnosis result was regarded as consistent with the overhaul results. The method proposed can be applied to erosion diagnosis of congeneric blast furnace hearth.