  A method is presented for calculating solutions to differential equations
analytically for a variety of problems in physics. An iteration procedure based
on the recently proposed BLUES (Beyond Linear Use of Equation Superposition)
function method is shown to converge for nonlinear ordinary differential
equations. Case studies are presented for solitary wave solutions of the
Camassa-Holm equation and for traveling wavefront solutions of the Burgers
equation, with source terms. The convergence of the analytical approximations
towards the numerically exact solution is exponentially rapid. In practice, the
zeroth-order approximation (a simple convolution) is already useful and the
first-order approximation is already accurate while still easy to calculate.
The type of nonlinearity can be chosen rather freely, which makes the method
generally applicable.
