Barycentric interpolation collocation methods for solving linear and nonlinear high-dimensional Fredholm integral equations.

In this article two barycentric interpolation collocation methods are proposed for solving linear and nonlinear high-dimensional Fredholm integral equations of the second kind. The approaches respectively utilize the modified weighted Lagrange functions and the novel rational functions as the interpolation basis functions. They are effective schemes for evaluating the multidimensional undetermined function. Through the numerical strategies and some composite quadrature formulas, the linear and nonlinear Fredholm integral equations are transformed into the corresponding linear and nonlinear algebraic equations. Further, we prove that the discrete collocation methods are equivalent to the Nyström quadrature methods. Then the convergence analysis is established by the collectively compact theory. Moreover, the error estimation of the approximate solution and the exact solution are also provided. Numerical examples are presented to illustrate the capability and efficiency of the techniques by compared with the classic Lagrange interpolation collocation method and other methodologies.