Advisor: Professor Emeritus Bshouty Daoud   Title: Computational Methods for Harmonic Mappings and its Uniqueness   Abstract; The well-known Riemann Mapping Theorem (RMT) for analytic mappings states the existence and uniqueness of conformal mappings between two simply connected domains. The existence of  RMT for harmonic mappings with specific dilatation holds whereas  the uniqueness is wide open. Given a harmonic mapping between the unit disk  and a polygonal domain, we determine its uniqueness using a computational method.