The one group two dimensional neutron diffusion equations has been solved by the Boundary element method. A program have been developed to solve neutron diffusion equation. Gauss quadrature method have been used for obtaining the coefficients of the fluxes and currents in this program. The test problem has one region and free source. The code has been written c code. It has been obtained boundary fluxes and currents. The Boundary element and Finite element methods are well suited for solving irregular geometrical problems, the Finite difference method is not suited. The BEM solves problems with infinite domains. Some simple test calculations are made.