  The Burgers' equation is a one-dimensional momentum equation for a Newtonian
fluid. The Cole-Hopf transformation solves the equation for a given initial and
boundary condition. However, in most cases the resulting integral equation can
only be solved numerically. In this work a new semi-analytic solving method is
introduced for analytic and bounded series solutions of the Burgers' equation.
It is demonstrated that a sequence transformation can split the non-linear
Burgers' equation into a sequence of linear diffusion equations. Each
consecutive sequence element can be solved recursively using the Green's
function method. The general solution to the Burgers' equation can therefore be
written as a recursive integral equation for any initial and boundary
condition. For a complex exponential function as initial condition we derive a
new analytic solution of the Burgers' equation in terms of the Bell
polynomials. The new solution converges absolutely and uniformly and matches a
numerical solution with arbitrary precision. The presented semi-analytic
solving method can be generalized to a larger class of non-linear partial
differential equations which we leave for future work.
