Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions

In this paper, we develop the orthogonal approximation by using generalized Laguerre functions. Some basic results on this approximation are established, which serve as the mathematical foundation of spectral methods for various differential equations on unbounded domains. As an example of applications, we propose a spectral method for a partial differential equation of degenerate type, which plays an important role in financial mathematics and other fields. The convergence of proposed scheme is proved. Numerical results show its spectral accuracy in space.