A capable collocation method for solving a class of nonlinear Volterra integral equations of the third kind

In this work, we propose a capable collocation method based on the Legendre polynomials to approximate the solution of Hamerestien third kind Volterra integral equations. The existence, uniqueness and regularity of the solution is discussed based on the nonlinear cordial Volterra integral operators. We use the properties of projection operators to prove that the approximate solution will be converged to the exact solution and the rates of convergence are obtained. Also, the structure and implementation of the method will be discussed. Moreover, some numerical examples are given to show the accuracy and efficiency of the method.