Abstract:

Let us consider a system of ODE's of the form F(x,y,y′,y″) = 0 where y and F are vector functions. By introducing an operator T such that Tu = F(x,u,u′,u″″) we have Ty = θ. Assuming that y° is an approximation of the solution y(x) a generalization of Newton's method can be applied to improve, under certain conditions, such approximation by the recursive algorithm yi+1=yi- T′-1 (y′) Ty(i = 0,1,2,...). In the present case we use such an approach in a numerical fashion as follows. After obtaining by any method of integration numerical approximations yn on a discrete set of points xn(n = 1,2,..., N) we interpolate them by a convenient function R(x). By taking this interpolant as the first analytical approximation Newton's process is applied pointwise in order to correct by iterations the discrete approximations yn. This procedure may become rapidly convergent especially in some stiff problems where we have obtained so far promissing results. © 1980.