Spline approximate solution for doubly periodic Riemann boundary value problem

摘要

A straightforward method for the approximate solution of doubly periodic Riemann boundary value problems for analytic functions is proposed through the approximation by the δ-cardinal splines of the first degree and the cubic δ-cardinal splines. First, we approximate the solution of the doubly periodic Riemann jump problem based on approximation of the singular integral operator with Weierstrass ζ-function kernel. Furthermore we obtain the approximate solution of the general non-homogenous doubly periodic Riemann problem. We prove that the approximate solution is sufficiently close to the exact solution in any degree when the partition Δ is sufficiently fine.