The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval − 1 , 1 is solved. The discontinuous solution on the domain − 1 , 1 is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.