The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis

We propose an effective method based on the reproducing kernel theory for nonlinear Volterra integro-differential equations of fractional order. Based on the Bernoulli polynomials bases, we construct some reproducing kernels of finite-dimensional reproducing kernel Hilbert spaces. Then, based on the constructed reproducing kernels, we develop an efficient method for solving the nonlinear Volterra integro-differential equations of fractional order. We reduce the nonlinear Volterra integro-differential equations of fractional order to the nonlinear Volterra integral equations of the second kind and use a simple iteration to overcome the nonlinearity of the problem. In fact, we are looking for an approximate solution to the problem in a finite-dimensional reproducing kernel Hilbert space. We investigate the convergence analysis of the proposed method thoroughly. Several numerical simulations are presented to demonstrate the performance of the method. The obtained numerical results confirm the theoretical results and show the efficiency and accuracy of the proposed method.