In the boundary element method (BEM), analytic fundamental solutions with weight functions are use as the solution of the field equation. Since these functions have singularity, when an integral point is close to the source point, the accuracy is determined by that of the numerical boundary integration. In the present papar, all integral equations for the potential and gradient, are obtained by a technique based on the uniform gradient condition. As a result, all of the singlarities in the integral equations for potential, gradient and these of the internal point are normalized in the same manner. Through some numerical results of models under several boundary conditions, it is shown that unknown nodal values along the boundary are more accurate than those obtained by the usual methods, and the accuracy of the potential and gradient at internal points near the boundary, has clearly been improved.