This approximation is ensured by the

This approximation is ensured by the uniform convergence of the flat series. The uniform convergence is proved in [3,4]. Similar function series (splitting expansions) are described in [5]; the concept of using the expansion in the form that it fak pathway is done in this paper comes from Schwartz and Huet [6]. The goal of the current paper is to obtain the approximate estimates for the a constant. This problem is solved by using the direct method based on calculating the distribution norms and distribution densities. Hereinafter we shall assume that the estimates are unfit if the calculation error of a is not less than 1 %.
The main properties of a boundary problem solution (1) As a first step of our study, let us examine the solution properties of an equation
Renormalization to a unit-length interval is not expedient in this case, as r = a–2/3 (the a and the r constants are connected). is of the following form:
The proof of Lemma 1 is obvious. It follows from Lemma 1 that and it is fairly easy to obtain a non-linear integral equation to determine h(ω). The distribution h(ω) ≥ 0 is monotonous, i.e. therefore, by virtue of the second law of the mean (see [7]):
Let us introduce a notation
or, in another form, □ Corollary to Lemma 2. An actual path (characteristic) of Eq. (3) satisfies the condition of the Z(1,ϕ,ψ) distribution minimum:
Z(h, ψ, ϕ) can be treated as an action, and instead of Crocco\'s Eq. (3) for the variable ϕ, the Jacobi equation for Z can be solved:
The full solution of the Jacobi equation is equivalent to the solution of Eq. (3). The inequality holds along the path of Eq. (3), where δ is an isometric variation of virtual trajectories, and d is a variation of a real-valued solution of the boundary problem (1). and allows for the following estimate:
The derivative in the left-hand side of the equality is nonnegative at
Blasius constant estimation (a) It follows from the proof of Lemma 1 that
It follows from (8) that . A similar approximation of the solution of boundary problem (1) is given in [8] where it is written as
By virtue of the equality (8), the constant a lies in the range 0.32573501 < a < 0.376126392. The σ estimate can be improved by using the (7) identity: with the obtained estimate being the upper bound for σm. Thus, which lies within the calculated deviation of a. (b) Suppose that in the general case,
Then formula (6) takes the form:
The arithmetic mean of α on the interval (0, 1/4) is 0.125; then with an error of 1% from the precise value. For σm = 0.7795, a = 0.33207, and the constant determination is narrowed down in error by 0.003 %. Thus, the Blasius constant can be obtained from the Crocco equation integral: assuming ϕ = 0, h = 1, σ = σm, σm > 3/4. Then we find . (c) Pseudo-theorem.The following rational estimate holds: a = 1/3. Such a flat series is described in a study by Varin [2], wherein the author uses C to denote the expansion parameter λ. Substituting the (3) series into the Crocco Eq. (1) with a free term f(h) splits the equation into a system of linear equations: and then
Obviously, this expansion does not depend on the parameter λ. Here
With h = 0, we find the equality or, in symbolic form, where . Then, using the formal approach, we find the equality
a = 1/3 = 0.3(3), The obtained value of a is somewhat different from the precise value found in [1,2], by no less than 32 digits, and is equal to 0.33205733621… , i.e. by less than 0.4 %. A splitting (flat) series for Eq. (3) leads to a sequence (system) of linear equations: each of them being a necessary condition for the respective distribution minima. Let us write out the equations in question in their natural order:
As a result, it turns out that the nonlinear boundary problem is “broken up” into a countable sequence of linear ones, each possessing an energy integral of the form