  The iteration sequence based on the BLUES (Beyond Linear Use of Equation
Superposition) function method for calculating analytic approximants to
solutions of nonlinear ordinary differential equations with sources is
elaborated upon. Diverse problems in physics are studied and approximate
analytic solutions are found. We first treat a damped driven nonlinear
oscillator and show that the method can correctly reproduce oscillatory
behaviour. Next, a fractional differential equation describing heat transfer in
a semi-infinite rod with Stefan-Boltzmann cooling is handled. In this case, a
detailed comparison is made with the Adomian decomposition method, the outcome
of which is favourable for the BLUES method. As a final problem, the Fisher
equation from population biology is dealt with. For all cases, it is shown that
the solutions converge exponentially fast to the numerically exact solution,
either globally or, for the Fisher problem, locally.
