The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind

This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynomials of the same degree, while the given potential function is approximated by monic Chebyshev polynomials of the same degree. Further, the two parametric functions associated to the parametrized kernel are expanded into Taylor polynomials of the first degree about the singular parameter, and an asymptotic expression is created, so that the obtained improper integrals of the integral operator become convergent integrals. Thus, and after using a set of collocation points, the required numerical solution is found to be equivalent to the solution of a linear system of algebraic equations. From the illustrated example, it turns out that the proposed method minimizes the computational time and gives a high order accuracy.