[Paper Review] Analytical approximation of Blasius' similarity solution with rigorous error bounds

Ovidiu Costin, S. Tanveer|arXiv (Cornell University)|Mar 6, 2013

Fractional Differential Equations Solutions7 references3 citations

TL;DR

This paper presents a rigorous analytical approximation for Blasius' similarity solution of the boundary layer equation using a hybrid method combining polynomial and error function representations. It achieves high-accuracy error bounds via contraction mapping and energy estimates, yielding f''(0) = 0.469600 ± 0.000022 with provable convergence and applicability to nonlinear ODEs beyond Blasius.

ABSTRACT

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 $\lim_{x ightarrow \infty} f^\prime (x) =1$. 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.

Motivation & Objective

• To develop a high-accuracy analytical approximation for the Blasius similarity solution with provable error bounds.
• To overcome the lack of rigorous error control in existing empirical approximations for nonlinear boundary value problems.
• To extend a perturbative method based on initial approximations to nonlinear ODEs without explicit small parameters.
• To establish a framework applicable to a broad class of linear and nonlinear differential equations with boundary conditions.

Proposed method

• Construct a piecewise approximation: a 12th-degree polynomial on [0, 5/2] and an error function-based expression on [5/2, ∞).
• Match the two representations at x = 5/2 to determine all free parameters, including f''(0).
• Use a contraction mapping argument in an adapted norm to control the residual error in the perturbation equation.
• Apply energy bounds for the Green’s function of the linearized problem to control error propagation.
• Employ Laplace transform techniques and asymptotic analysis to estimate remainder terms in the approximation.
• Use explicit integral representations and inequalities to bound the remainder functions R3 and R4, ensuring uniform error control.

Experimental results

Research questions

• RQ1 Can a high-accuracy analytical approximation be constructed for Blasius’ equation with rigorous error bounds?
• RQ2 How can the absence of a small parameter in Blasius’ problem be overcome to apply perturbative methods?
• RQ3 What is the optimal way to match local polynomial and global special function approximations in a boundary value problem?
• RQ4 Can energy-based estimates replace detailed Green’s function analysis for error control in nonlinear ODEs?
• RQ5 What is the tightest provable bound for f''(0) in the Blasius problem using this framework?

Key findings

• The method produces a piecewise analytical approximation: a 12th-order polynomial on [0, 5/2] and an error function expression on [5/2, ∞), matched at x = 5/2.
• The wall shear stress is computed as f''(0) = 0.469600 ± 0.000022, with rigorous error bounds derived from contraction mapping and energy estimates.
• The remainder terms R3 and R4 are bounded by R3,m ≤ 0.02057 and R4,m ≤ 0.009042 for t ≥ 1.99, ensuring convergence.
• The approach is generalizable to a wide class of nonlinear and linear differential equations with initial or boundary conditions.
• The method replaces detailed Green’s function analysis with softer energy-based estimates, simplifying the error control framework.
• The final error bound for the solution is controlled to within 1.02×10⁻⁸ for t ≥ T, demonstrating high numerical reliability.