Numerical solution of singular Fredholm integral equations of the first kind using Newton interpolation

In this paper a computational technique is presented for the numerical solution of a certain potential-type singular Fredholm integral equation of the first kind with singular unknown density function, and a weakly singular logarithmic kernel. This equation is equivalent to the solution of the Dirichlet boundary value problem for Laplace equation for an open contour in the plane. The parameterization of the open contour facilitates the treatment of the density function’s singularity in the neighborhood of the end-points of the contour, and the kernel’s singularity. The unknown density function is replaced by a product of two functions; the first explicitly expresses the bad behavior of the density function, while the second is a regular unknown function, which will be interpolated using Newton interpolation in a matrix form. The singularity of the parameterized kernel is treated by expanding the two argument parametric functions into Taylor polynomial of the first degree about the singular parameter. Moreover, two asymptote formulas are used for the approximation of the kernel. In addition, an adaptive Gauss–Legendre formula, is applied for the computations of the obtained convergent integrals. Thus the required numerical solution is found to be equivalent to the solution of a system of algebraic equations. The numerical solution of the illustrated example is closer to the exact solution; which ensures the high accuracy of the presented computational technique.