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Boundary elements have emerged as a powerful alternative to finite elements particularly in cases where better accuracy is required. The most important features of boundary elements however is that it only requires descretization of the surface rather than the volume. Here, A general algorithm of the boundary integral method has been formulated for solving elliptic partial differential equations. The broad applicability of the approach is illustrated with a problem of practical interest giving the solution of the Laplace equation for potential flow with mixed boundary problems. The results and patterns are shown in tables and figures and compared well with Brebbia [1] are found in good agreement.