Abstract In this study the Fredholm integral equations of the first and the second kind have been examined using two numerical approaches: “adding and subtracting of the singularity” and “piecewise linear panels” methods. While, the solution for the Fredholm integral equation of the first kind (kinematic condition-no flux) shows good agreement with the analytical solution, the Fredholm integral equation of the second kind generates spurious results only in the case of the piecewise linear panels method. It was found that the spurious solutions are a direct result of the low order of numerical scheme accuracy in the piecewise linear panels method. In order to correct the structure of the difference scheme in the piecewise linear panels method, the numerical error was redistributed to preserve the conservative form of the circulation at the difference scheme level. The obtained solution in the conservative form does not exhibit any spurious results. This has direct consequences on a moving boundary such as a free interface, where it is shown that the non conservative form of the “piecewise linear panels” difference scheme is exhibiting spurious results, while the conservative form has a good agreement with the analytical solution.