On the method of solving nonlinear Fredholm integral equation of the second kind with piecewise-smooth kernels

The work is devoted to the development of iterative methods for solving nonlinear Fredholm integral equations of the second kind with piecewise-smooth kernels. A new approach to constructing their solutions is proposed which combines the method of successive approximations with polynomial interpolations of functions on the segment $[-1,\,1]$. In this case, the original integral equation is reduced to a Volterra-type equation where the unknown function is defined on the segment mentioned. The free term of the equation is chosen as the initial approximation. At each iteration of  successive approximations method, the kernel of the integral equation is represented as a partial sum of a series in Chebyshev polynomials orthogonal on the segment $[-1,\,1]$. The coefficients in this expansion are found using the orthogonality of vectors formed by the values of these polynomials at the zeros of the polynomial whose degree is equal to the number of unknown coefficients. An approximation of the solution is made by interpolation of the obtained values of the solution at the Chebyshev nodes at each iteration. The work also constructs a solution to an integral equation whose free term has a discontinuity point of the first kind. The results of the computational experiments are presented, which demonstrate the effectiveness of the proposed approach.