Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this work, we construct numerical solutions to a boundary value problem defined on a spherical cap using radial basis functions. The error analysis between the exact solution and the approximation is provided. Numerical experiments are presented to confirm theoretical estimates. This is a joint work with Kerstin Hesse.