Abstract

With the conformal map from unit square to unit disk analytically known, in our last contribution we investigated ways to numerically map the disk to more general (but star-shaped) domains. Such point-mappings of the complex planes are now to be interpreted as transformations of co-ordinates, hence the domains are parametrized by the square. Using general, curvilinear co-ordinates one has to take the shape of the fundamental tensor and other related quantities into account. The also numerically known derivatives of the map act as metric quantities flowing in and correcting a pre-given PDE in Cartesian co-ordinates on such a domain. On the other hand, a formulation of physical laws in co-ordinate free manner gives an even smarter access to implement a simulation code of a given problem in Mathematica. In this article, we focus on a practical problem: let the domain be the cross-section of a tooth and the task be to find the temperature distribution on its boundary with respect to a heat source moving in the interior of the domain. This model can then be interpreted as a decision-finding issue to parameter identification when treating a tooth with a laser pulse. Considering the problem in three dimensions by using rotational symmetry will turn out to be essential with respect to the obtained results.