Note on the solution of a certain boundary-value problem

Abstract

A simple algorithm is described for inverting the operatorD _x D _y (D _x andD _y here and subsequently denote partial differentiation with respect tox andy respectively) which occurs in the iterative solution of the equationD _x D _y f (x, y)=g (x, y, f, D _x f, D _x ² f,D _x D _y f, D _y ² f) when boundary values off(x,y) are given along the sides of the rectangle in thexy-plane whose corners are at the points (a,b); (a+(n+1)k,b); (a+(n+1)k,b+(n+1)h); (a,b+(n+1)h).