Matrix-Vector Formulas of the Barycentric Lagrange Interpolation for Solving Systems of Two Linear Fredholm Integral Equations of the Second Kind

In this study, we provided a new approach to solve systems of second-kind linear Fredholm equations. The presented method is based on the application of some new formulas of the barycentric Lagrange interpolation via matrices. By applying these new formulas, we were able to express the unknown functions through four matrices, one of which is the monomial basis matrix. As for the given kernels, we have been able to express each of them as a product of five matrices, two of which are monomial basis. We obtained a linear algebraic system in the form of block matrices by inserting the two interpolated unknown functions and the interpolated four kernels into the two equations of the system. The solution of this block-matrix system produced the values to the two unknown coefficients matrices of the two unknown functions. We solved eight examples, where we got numerical solutions that are equal to the exact ones if the two unknown functions and the four given kernels are algebraic functions. The solutions that we obtained converge strongly with the exact solutions in the case that the data function is a singular function, which clearly demonstrated the superiority of the proposed method over other cited methods.