On the finite difference scheme for a non-linear evolution problem with a non-linear dynamic boundary condition

We derive a 1-D explicit parabolic finite difference scheme for a non-linear evolution problem with a non-linear dynamic boundary condition. The derivation starts with the problem as an implicit evolution equation; hence an abstract Cauchy problem. The abstract Cauchy problem is associated with canonical operators of the implicit evolution equation, whose range is a product Hilbert space of two Sobolev spaces. A parabolic finite difference scheme for the implicit evolution equation (on the product Hilbert space), is derived. This scheme, together with the finite difference scheme for the boundary surface heat loss relation is used to obtain a scheme that calculates both the body temperature and the "corresponding" surface temperature in the heat loss through radiation problem.