A Modification of the Shooting Method for Two-Point Boundary Value Problems

Abstract

The well-known shooting method for the numerical solution of the two-point boundary value problem y″ = f(x,y,y′), y(a) = A, y(b) = B, is sometimes referred to as “the garden hose method”. Suppose one holds the nozzle of a garden hose at the point (a, A) with the object of hitting, with the jet of water, a distant target at (b,B). Trial and error attempts to establish the correct angle at which to point the nozzle then constitute an analogue of the conventional shooting method. There is, however, another practical possibility. One can walk right up to the target, point the nozzle directly at it, and then retreat to the point (a,A), meanwhile continuously altering the angle of the nozzle so that the water continues to play on the target. This is the analogue of the method to be described. Note that it is not necessary to retreat along the final trajectory of the jet! One can retreat along any path joining (b,B) to (a,A).