Two numerical treatments for solving the linear integro-differential Fredholm equation with a weakly singular kernel

We compare the error behavior of two methods used to find a numerical solution of the linear integro-differential Fredholm equation with a weakly singular kernel in Banach space $C^1[a,b]$. We construct an approximation solution based on the modified cubic $b$-spline collocation method. Another estimation of the exact solution, constructed by applying the numerical process of product and quadrature integration, is considered as well. Two proposed methods lead to solving a linear algebraic system. The stability and convergence of the cubic $b$-spline collocation estimate is proved. We test these methods on the concrete examples and compare the numerical results with the exact solution to show the efficiency and simplicity of the modified collocation method.