Approximate Solutions to Weakly Singular Fredholm Integral Equations of the Second Kind using Shifted Legendre Polynomials of the First Kind

In this research, we presented a simple approach to approximate the second type of linear weakly singular and non-singular Fredholm integral. Shifted Legendre polynomials of the first kind in matrix-vector forms were used to construct the approach. The singularity of the kernel was removed analytically. Theorems regarding the convergence of the estimations of the error norm and the mean were proved. The numerical examples demonstrated the method's uniqueness and precision.