**6. Conclusion**

*<sup>L</sup>*<sup>≈</sup> � *<sup>λ</sup>*<sup>2</sup>

*Mathematical Theorems - Boundary Value Problems and Approximations*

results.

**Table 1.**

**Table 2.**

**Table 3.**

**92**

for any value of used parameter.

*<sup>μ</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*14*μ*<sup>2</sup>

Expression (60) has a pole at *μ* ¼ �12*:*66. Below, one can see some numerical

*μ* **L (4) L (NR)** 1.057 1.0648 1.513 1.4724 1.685 1.6358 1.759 1.7398 1.768 1.7691

*Comparison for L of numerical results (NR) from paper by [33] with TPPA formulate (60).*

*M T***<sup>s</sup>** *ς<sup>m</sup> T***<sup>1</sup>** 3 0.426 1.340 3 0.744 9.127 5 0.637 7.929 7 0.531 6.676 3 0.806 22.00 5 0.756 20.82 7 0.712 19.75

*TPPAs parameters for different Mach numbers* M*, temperature* T*S, and* n *= 1 values.*

*TPPAs parameters for different Mach numbers* M*, temperature* T*S, and* n *= 0.76 values.*

*M* **5 10 10** *T*<sup>s</sup> 335 *a*<sup>2</sup> 0.17 0.74 0.56 *ς<sup>m</sup>* 0.729 0.714 0.798 *c* 1.45 1.42 1.38 *T*<sup>1</sup> 0.56 8.69 8.16

In **Table 1**, the second column is made by calculation results by formula (4), the third column is made by paper data [33]. One can see that TPPA gives good result

In Section 4.4, the problem was solved for several variants of the Mach number and the heating temperature: *M* ¼ 5; 10; 15, *Ts* ¼ 3; 5; 7 of the streamlined flat plate, with constant Prandtl number values *σ* ¼ 0*:*76, adiabatic index *κ* ¼ 1*:*4, and two values of dynamical viscosity index *<sup>μ</sup>* <sup>¼</sup> *<sup>T</sup><sup>n</sup>* : *<sup>n</sup>* <sup>¼</sup> 1; 0*:*76. When the first equation of the systems (43) and (44) is solved, it becomes independent of the second equation and can be compared with the known Blasius solution (see Section 3), which was used as a test when compared to our method [35–40]. Thus, the value of

<sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*079*μ*<sup>2</sup> (60)

The procedure of constructing the PA is much less labor-intensive than the construction of higher approximations of perturbation theory. PA can be applied to power series but also to the series of orthogonal polynomials. PA is locally the best rational approximation of a given power series. They are constructed directly and allow for efficient analytic continuation of the series outside its circle of convergence, and their poles in a certain sense localize the singular points (including the poles and their multiplicities) of the function at the corresponding region of convergence and on its boundary. PA is fundamentally different from rational approximations with (fully or partially) fixed poles, including the polynomial approximation, when all the poles are fixed in infinity. That is the above property of PA—effectively solving the problem of analytic continuation of power series—lies at the basis of their many successful applications in the analysis and the study of applied problems. Currently, the PA method is one of the most promising nonlinear methods of summation of power series and the localization of its singular points. Including the reason why the theory of the PA turned into a completely independent section of approximation theory, and these approximations have found a variety of applications both directly in the theory of rational approximations, and in perturbation theory.

Thus, the main advantages of PA compared with the Taylor series are as follows:


3.PA can establish the position of singularities of the function.

TPPA allows to overcome the locality of asymptotic expansions, using only a few terms of asymptotics. Unfortunately, the situations when both asymptotic limits

have the form of power expansions are rarely encountered in practice, so we have to resort to other methods of AEFs construction, for example, the method quasirational approximation which is described in [23]. The method of combination (combining method) of asymptotics by using TPPA is alternative to the well-known matching method [6]; it is useful in local domains of transition layers where asymptotics are not uniform. This method was tested on well-known problems of mathematical physics, in particular, problems of fluid dynamics. The main advantage of the method is that it has an analytic form.

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