We propose a new boundary integral equation method (BIEM) to model the spontaneous propagation of rupture on a planar fault embedded in a homogeneous elastic medium. The BIEM formulation is very compact and fast in computation, so that we are able to study the effect of different slip- and rate-dependent friction laws on dynamic shear fault propagation. We simulated a spontaneous rupture by a sudden break of asperity under two friction laws, rate and/or slip weakening. We examined both long and circular asperities. The long asperity model corresponds to the 2D anti-plane or in-plane problem. The obtained result shows that slip-weakening friction is important at the crack tip, and rate-weakening friction plays an important role in the healing stage. If slip-weakening friction is strong stress drops gradually at the crack tip. On the other hand, if rate-weakening friction is strong stress drops abruptly, but it stops suddenly and sometimes stress recovers. This sudden stop of rupture produces a heterogeneous stress distribution, which will in turn produce aftershocks. Finally, we studied a realistic asperity model in which rupture starts from a small patch and then propagates with finite rupture speed. Depending on the asperity size healing can occur even without rate-weakening friction. This is due to very strong healing waves produced by the edges of the asperity.