Approximation methods for the Muskhelishvili equation on smooth curves,

We investigate the possibility of applying approximation methods to the famous Muskhelishvili equation on a simple closed smooth curve $\Gamma $. Since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way. It is shown that the spline Galerkin, spline collocation and spline qualocation methods for the corrected equation are stable and the corresponding approximate solutions converge to an exact solution of the Muskhelishvili equation in appropriate norms. Numerical experiments confirm the effectiveness of the proposed methods.