Numerical approximation of Fredholm integral equation by the constrained mock-Chebyshev least squares operator

In this paper, we propose two numerical approaches for approximating the solution of the following kind of integral equation $f(y)−\mu\int_{-1}^{1}f(x)k(x,y)w(x)dx=g(y),\,y∈[−1,1]$, where $f$ is the unknown solution, $\mu \in \mathbb{R} \smallsetminus \{0\}$, $k,g$ are given functions not necessarily known in the analytical form, and $w$ is a Jacobi weight. The proposed projection methods are based on the constrained mock-Chebyshev least squares polynomials, and starting from data known at equally spaced points, provide a fine approximation of the solution. Such peculiarity can be helpful in all cases we deal with experimental data, typically measured at equispaced points. We prove the introduced methods are stable and convergent in some Sobolev subspace of C[−1,1]. Several numerical tests confirm the theoretical estimates and numerical effectiveness of the proposed methods.