Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels

This paper proposes a quadrature method based on multi-variate Bernstein polynomials. The method is used to solve multidimensional Volterra integral equations with weakly singular kernels. Firstly, we use multi-variate Bernstein polynomials to approximate the unknown function of an equation, then a discrete function equation can be obtained by substituting the approximate solution into the equation. Secondly, the discrete function system is transformed into an algebra equation system by using some discrete points. We can perform the integral operations without discrete kernel function, and the weakly singular integrals can be calculated directly by using quadrature method, so the method is easy to implement. Thirdly, we prove the existence and uniqueness of the solution of the approximate equation, as well as the error analysis of the proposed method. Six numerical examples are given to illustrate the efficiency of this method.