Solving the Inverse Cauchy Problem of the Laplace Equation Using the Method of Fundamental Solutions and the Exponentially Convergent Scalar Homotopy Algorithm (Ecsha)

In this paper, the inverse Cauchy problem of the Laplace equation is considered. Using the method of fundamental solutions, a system of linear algebraic equations can be obtained by satisfying the Cauchy boundary conditions on the overprescribe boundary points. The resulting linear algebraic equation is ill-posed and is treated by the exponentially convergent scalar homotopy algorithm (ECSHA). Four examples are adopted to show the validity of the proposed numerical scheme and it is concluded that the current approach can successfully resolve the ill-posedness of the inverse Cauchy problem even when the noise exists.

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