Approximation of the double-layer potential solution on the open boundary curves

Abstract

In the Dirichlet problem with a smooth open boundary curve in $R^2$, the double-layer potential is introduced as a solution, u(P). In order for u(P) to be a continuous solution in $R^2$, precisely through the boundary curve S, it should satisfy a "jump relation" resulted from discontinuity of the kernel, k through S. Now, the problem is converted to finding the density function, g(P). To obtain the density function g(P), the jump relation is formulated explicitly as the form of Fredholm``s integral equation in this thesis. If $g_n$ satisfying the jump relation is obtained in a proper subspace $X_n$ of C ($R^2$), then $u_n$(P) including $g_n$ in it``s integral formula is an approximation to the exact solution, u(P) of the Dirichlet problem given above.