Numerical Solution of Dirichlet Boundary Domain Integro Differential Equation with Less Number of Collocation Points

In this paper, we show that we have two approaches in implementing of Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet Boundary Value Problem (BVP) for an elliptic Partial Differential Equation (PDE) with a variable coefficient. One way is by choosing the collocation points at all nodes i.e. on the boundary and interior domain. The other approach is choosing the collocation points for the interior nodes only. We present the numerical implementation of the BDIDE associated to Dirichlet BVP for an elliptic PDE with a variable coefficient by using the second approach. The BDIDE is consisting of several integrals that exhibit singularities. Generally, the integrals are evaluated by using Gauss- Legendre quadrature formula. Our numerical results show that the use of semi-analytic method gives high accuracy results. The discretized BDIDE yields a system of equations. We then apply by a direct method i.e. LU decomposition method to solve the systems of equations. In all the test domains, we present the relative errors of the solutions and the relative error for the gradient.