Bernstein numerical method for solving nonlinear fractional and weakly singular Volterra integral equations of the second kind

An iterative numerical method is proposed for solving nonlinear fractional Volterra integral equations. This method is based on Picard iterations and at each iterative step we will apply a piecewise Bernstein polynomial approximation technique. The convergence of the method is proved by providing the error estimate in the discrete and continuous approximation. The theoretical results are tested on some numerical examples, illustrating the performances of the proposed method.