  We use a recently developed method \cite{Costinetal}, \cite{Dubrovin} to find
accurate analytic approximations with rigorous error bounds for the classic
similarity solution of Blasius of the boundary layer equation in fluid
mechanics, the two point boundary value problem $f^{\prime \prime \prime} + f
f^{\prime \prime} =0$ with $f(0)=f^\prime (0)=0$ and $\lim_{x \rightarrow
\infty} f^\prime (x) =1$. The approximation is given in terms of a polynomial
in $[0, \frac{5}{2}]$ and in terms of the error function in $[\frac{5}{2},
\infty)$. The two representations for the solution in different domains match
at $x=\frac{5}{2}$ determining all free parameters in the problem, in
particular $f^{\prime \prime} (0) =0.469600 \pm 0.000022 $ at the wall The
method can in principle provide approximations to any desired accuracy for this
or wide classes of linear or nonlinear differential equations with initial or
boundary value conditions. The analysis relies on controlling the errors in the
approximation through contraction mapping arguments, using energy bounds for
the Green's function of the linearized problem.
