A generalized quasi‐boundary value method for the backward semi‐linear time‐fractional heat problem in a cylinder

We are concerned with a backward problem associated with a semi‐linear time‐fractional heat equation in an axis‐symmetric cylinder, which arises from the modeling of the blast furnace steelmaking in metallurgy. Under some assumptions, the existence and uniqueness of the solution to the semi‐linear problem is first established. The ill‐posedness of the backward problem is then established, and we obtain the error estimates by a generalized quasi‐boundary value regularization method. Finally, the numerical experiment is presented to demonstrate the effectiveness of the proposed method.