  A new (algebraic) approximation scheme to find {\sl global} solutions of two
point boundary value problems of ordinary differential equations (ODE's) is
presented. The method is applicable for both linear and nonlinear (coupled)
ODE's whose solutions are analytic near one of the boundary points. It is based
on replacing the original ODE's by a sequence of auxiliary first order
polynomial ODE's with constant coefficients. The coefficients in the auxiliary
ODE's are uniquely determined from the local behaviour of the solution in the
neighbourhood of one of the boundary points. To obtain the parameters of the
global (connecting) solutions analytic at one of the boundary points, reduces
to find the appropriate zeros of algebraic equations. The power of the method
is illustrated by computing the approximate values of the ``connecting
parameters'' for a number of nonlinear ODE's arising in various problems in
field theory. We treat in particular the static and rotationally symmetric
global vortex, the skyrmion, the Nielsen-Olesen vortex, as well as the 't
Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the
monopole is also computed by the new method. We also consider some ODE's coming
from the exact renormalization group. The ground state energy level of the
anharmonic oscillator is also computed for arbitrary coupling strengths with
good precision.
