A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation

We consider an inverse problem for Laplace equation by recovering the boundary value on an inaccessible part of a circle from an overdetermined data on an accessible part of that circle. The available data are assumed to have a Fourier expansion, and thus the finite terms truncation plays a role of regularization to perturb the ill-posedness of this inverse problem into a well-posed one. Hence, we can apply a modified indirect Trefftz method to solve this problem and then a simple collocation technique is used to determine the unknown coefficients, which is named a modified collocation Trefftz method. The results may be useful to detect the corrosion inside a pipe through the measurements on a partial boundary. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown data from the given data under noise.