On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x, y, u, ux, uy)

Abstract

This paper1 develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x0)=y0) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation uxy=f(x, y, u, ux, yy), u(x, y0)=σ(x), u(x0, y)=τ(y), where σ(x0)=τ(y0). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions σ(x) and τ(y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.