On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.