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

Analytical approximation of Blasius' similarity solution with rigorous error bounds