Finite difference methods for a singular cauchy problem

Abstract

The Cauchy problem for the equationh(x, y)y2a w_yy_w x _x: = (*,y), (i)with initial conditionsW(x, 0) - d(x), Wy(x, 0) = C(x), (2)is solved using finite difference methods. We assume α is any positive real number and that h(x, y) is positive on the closure of the domain under consideration.We reduce (1), (2) to a first order system of integral equations. Introducing a characteristic mesh and an appropriate difference approximation for this system of integral equations, convergence is obtained in a neighborhood of the initial segment.