  This paper investigate on numerical analysis on modified Single-layer
approach to exterior Dirichlet problem of Laplace equation. We complete the
convergence and error analysis of Petrov-Galerkin and Galerkin-Collocation
methods with trigonometric basis for the induced modified Symm's integral
equation of the first kind on analytic boundary. Besides, utilizing the
composite trapezial quadrature formula and trigonometric interpolation to
handle the singularity in modified logarithmic kernel, we establish the
numerical procedure for implementation. On these numerical examples, we compare
the effect and efficiency of different Petrov-Galerkin and Galerkin-Collocation
methods.
