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br Introduction For Crocco s equation given on the
Introduction
For Crocco\'s equation
given on the interval 0 < 1, with f(h) >0, f ∈ L1(0, 1), it SAR 405 is possible to set various boundary problems.
First of all, a typical boundary problem can be set:
Aside from this one, a boundary problem homogeneous in φ(h) can be set:
as well as a combined homogeneous boundary problem:
with real parameters 0, 1.
The boundary problems (1)–(4) have a physical basis and are connected to describing the phenomena of diffusion, thermal conductivity, and the jet and the wall viscous boundary layers.
Ref. [1] suggests using the Cauchy conditions:
instead of the setting (2) when solving a typical Crocco boundary problem for the case f(h)=h, and the constant α can be adjusted so that φ(1)=0.
Let φ=φ(h,α) and φ(1,α)=0. Then
It should be noted that this assertion is none other than the theorem on the continuous dependence of the solution of a differential equation on the parameters [2].
Ref. [1] proved that the point h = 1 is a movable singularity of the Cauchy problem (1), (2а):
[1,3], but .
The analysis of the existing results
The analysis of Ref. [1] shows that a wider statement has actually been proved, i.e., that the convergence radius of a plane series for the dφ/dh derivative is equal to unity [1,3,4].
This result can be made more specific. Let α >0 be such a value of φ(0) for which φ(1, α)=0 and β >0 (β is any random value of φ(0), such that φ(h0)=0). Then
For a complete coincidence with the Crocco boundary problem, let us set r=α–2/3, or α=r–3/2, and, consequently, an increase in α leads to a decrease in the convergence radius r, and vice versa. Thus, we can state that
Then it is easy to calculate that
In order to carry out the adjustment, let us use the fixed-point technique and implement the Cauchy sequence; the completion of a normalized ring C1 ensures the convergence (this technique is known in its linearized form [5]).
Let
where the lower index is the iteration number.
Evidently, this equation is the iterative counterpart of Eq. (1). Then
is the iterated solution of the Cauchy problem (1), (2а). Therefore,
Let us further assume that f(h)=h. Then if under the zero-order approximation φ0(h)=α, then φ1(h)=(1 – h3)/12. In the first approximation, we obtain that
which is by 13% less than the corresponding exact value.
By writing this solution in the form
we obtain that
It follows that the value of φ(h, α) near the right endpoint of the interval h∈(0, 1) does not decrease with an increase in α, which goes to explain the adjustment algorithm.
In the second approximation,
which is higher by 30% than the exact value. The mean value of the α constant (its relaxation) over the first two approximations is 0.3514, which produces an error as large as 6%.
In the boundary problem (1), (3) the point h = 1–0 is a singularity for the derivative, namely, for h→1 –0, dφ/dh→-∝.
It can be proved (and we intend to do this further on) that the occurring derivative singularity is of logarithmic type, and . Therefore, the function φ2(h) is regular on the interval 0