An approximation method for the hypersingular heat operator equation

The original model problem is the two-dimensional heat conduction problem with vanishing initial data and a given Neumann-type boundary condition. In particular, certain choices of the representation formula for the heat potential yield the hypersingular heat operator equation of the first kind. In this paper we concentrate on the problem of solving this hypersingular integral equation. Our approximation method is a Petrov–Galerkin method, where we use collocation with respect to the space variable and Galerkin method with respect to the time variable. The trial functions are tensor products of piecewise cubic (space) and piecewise linear (time) smoothest splines. Stability and convergence of the resulting scheme is proved when the spatial domain of the original heat conduction problem is a disc.