Error Estimates of Finite Difference Methods for the Fractional Poisson Equation with Extended Nonhomogeneous Boundary Conditions

Abstract

Two efficient finite difference methods for the fractional Poisson equation involving the integral fractional Laplacian with extended nonhomogeneous boundary conditions are developed and analyzed. The first one uses appropriate numerical quadratures to handle extended nonhomogeneous boundary conditions and weighted trapezoidal rule with a splitting parameter to approximate the hypersingular integral in the fractional Laplacian. It is proven that the method converges with the second-order accuracy provided that the exact solution is sufficiently smooth and a splitting parameter is suitably chosen. Secondly, if numerical quadratures fail, we employ a truncated based method. Under specific conditions, the convergence rate of this method is optimal, as error estimates show. Numerical experiments are provided to gauge the performance of the methods proposed.