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Throughout this discussion we shall assume sufficient regularity conditions for /, a, and t so as to insure that equations (4.1), (5), (6) have meaning in any context in which they are used. We also assume that values of ux(0, y) and uy(x, 0) as well as a (x) and r (x) have been calculated: techniques on how this may be done are discussed in [5], In order to apply quadrature methods for the numerical solution of the above initial value problem, we convert the differential equation (1) into a system of integral equations.

A Gaussian quadrature method for the numerical solution of the characteristic intitial value problem 𝑢_{𝑥𝑦}=𝑓(𝑥,𝑦,𝑢)