**5. Results**

We complement the last equalities (50) and (51) with a normalizing condition:

∞ð

*<sup>a</sup>*ð Þ*ζ dζ* (49)

*<sup>B</sup>* (51)

ð Þ *ζ<sup>m</sup>* (52)

� � � � (53)

*<sup>m</sup>* � *ζ<sup>m</sup>*

(50)

(54)

0 *T*0

Following the procedure of the previous section, we will calculate the coefficients at *ζ* and *ζ*<sup>2</sup> in asymptotic expansions (44) and, equating them with the corresponding expressions from Eqs. (46) and (47), we will obtain equalities, from

1 ¼ *Ts* þ

which values *α*1, *α*2, *α*4, *β*0, *β*<sup>1</sup> are expressed through *a*1,*c*, *T*1, *ζm*:

*Mathematical Theorems - Boundary Value Problems and Approximations*

*T*1 *Ts*

*ζm T*1 � 1 *T*1

� 1 þ

Three parameters in asymptotics (44) are defined in the outer region if the

*β*<sup>1</sup> ¼ �<sup>1</sup>*=*

A priori at large *M* numbers, it is known that the temperature profile is non-monotonic and has a maximum within the layer at point *ς<sup>m</sup>* at which, as can be seen from the second equation of the systems (40) and (41), the following

*<sup>T</sup>*00ð Þ¼� *<sup>ζ</sup><sup>m</sup> <sup>a</sup>σφ*00<sup>2</sup>

From the convexity condition of the temperature profile in the vicinity of the

*<sup>a</sup>*ð Þ¼ *<sup>ζ</sup><sup>m</sup>* exp �*σ ζ*<sup>2</sup>

*<sup>a</sup>* 2 � *φ*<sup>0</sup> ð Þ*dζ*

Let us add the received equations with the integrated ratios received on the basis of coincidence of TPPAs (46) and (47); in this case, three members in asymptotic decompositions (50) and (51), the initial system of Eqs. (40) and (41), with boundary conditions (42) and (43), by using the technique stated in the previous

*c*2

<sup>þ</sup> ð Þ *<sup>n</sup>* � <sup>1</sup> *<sup>ζ</sup><sup>m</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*2ð Þ *<sup>a</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>* , *<sup>α</sup>*<sup>4</sup> <sup>¼</sup> 4,

*T*1 2

ð Þ *n* � 1 *a*<sup>2</sup>

, *β*<sup>1</sup> ¼ *σc*

*α*<sup>1</sup> ¼ *a*<sup>2</sup> þ *c*,

*<sup>α</sup>*<sup>2</sup> ¼ � <sup>1</sup> 2

*<sup>β</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ζ</sup><sup>m</sup> T*1

point *ςm*, the following equality is used:

*<sup>β</sup>*<sup>0</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þ *<sup>ζ</sup><sup>m</sup> <sup>a</sup>σφ*00<sup>2</sup>

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2 ∞ð

*c* ¼ ∞ð

0

0 *φ*0

2 � *φ*<sup>0</sup> ð Þ*dζ*

The integral relation for parameter *A* is obtained by multiplying Eq. (40) by

exp *<sup>ζ</sup>*<sup>2</sup> � *<sup>c</sup><sup>ζ</sup>* � �

following condition is met:

condition is used:

sections.

**90**

As an example of TPPA (see Section 3.2) used for matching of limiting asymptotics, consider the paper by Grasman et al. [33]. They dealt with Lyapunov exponents which characterize the dynamics of a system near its attractor. For the Van der Pol oscillator:

$$
\ddot{\mathbf{x}} + \mu \dot{\mathbf{x}} (\mathbf{x}^2 - \mathbf{1}) + \mathbf{x} = \mathbf{0} \tag{57}
$$

Similar to the asymptotic approximation of amplitude and period, expressions are derived for the nonzero Lyapunov exponent *λ*<sup>2</sup> for both small and large parameter *μ* values:

$$
\lambda\_2 = -\mu - \frac{1}{16}\mu^3 + \frac{263}{18432}\mu^5 + \dots, \mu \to 0,\tag{58}
$$

$$\lambda\_2 = -\frac{\mathfrak{Z} + 4\ln 2}{2(\mathfrak{Z} - 2\ln 2)}\mu + \dots, \mu \to \infty. \tag{59}$$

The overlap of these series does not take place. The authors of [33] remark: "Such an overlap comes within reach if in the regular expansion a large number of terms is included." This is not correct, because the obtained series is asymptotic; so, with increasing of number of terms, the results will be worst. So, one needs a summation procedure. Some authors [34] proposed to use PAs, but in this case one needs hundreds of perturbation series terms. That is why we use TPPA. Using two terms from expansion (58) and one term from expansion (59), one obtains

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$L \approx -\frac{\lambda\_2}{\mu} = \frac{1 + 0.14\mu^2}{1 + 0.079\mu^2} \tag{60}$$

the parameters according to the exact solution is equal: *a*<sup>2</sup> ¼ 0*:*664; *c* ¼ 1*:*72. Our decision gives *a*<sup>2</sup> ¼ 0*:*6641; *c* ¼ 1*:*7308. Of course, such a good match is due to the fact that these parameters are largely determined by local internal asymptotics, more precisely, derived from the function on the wall. But also within the transition area, the deviation from the exact solution does not exceed 1÷2% (for *φ*<sup>0</sup> and *T*, respectively). Design values of parameters for determining approximations (37)

*Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer…*

If *n* ¼ 0*:*76, this value corresponds to the physical characteristics of the air, and the constant calculation results for the approximation formulas (49) and (50) are

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

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

1.Typically, the rate of convergence of rational approximations greatly exceeds the rate of convergence of polynomial approximation. For example, the function e*<sup>ε</sup>* in the circle of convergence approximated by rational polynomials *Pn*(*ε*)*/Qn*(*ε*) in 4*<sup>п</sup>* times better than an algebraic polynomial of degree 2*n*. More tangible, it is property for functions of limited smoothness. Thus, the function |*ε*| on the interval [�1,1] cannot be approximated by algebraic polynomials so that the order of approximation was better than 1/*n*, where *n* is the degree of

<sup>2</sup>*<sup>n</sup>* � � <sup>p</sup> .

polynomial. PA gives the rate of convergence � exp � ffiffiffiffiffi

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

2.Typically, the radius of convergence of rational approximation is large compared with the power series. Thus, for the function arctan ð Þ *x* , Taylor polynomials converge only if j j *ε* ≤1, and PA is everywhere in *С*\((*- i*∞, - *i*]

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

and (38) for *n* = 1 are given in **Table 2**.

*DOI: http://dx.doi.org/10.5772/intechopen.93084*

shown in **Table 3**.

**6. Conclusion**

perturbation theory.

∪ [*i, i*∞)).

**93**

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

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 for any value of used parameter.


**Table 1.**

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

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


**Table 2.** *TPPAs parameters for different Mach numbers* M*, temperature* T*S, and* n *= 1 values.*


**Table 3.**

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

*Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer… DOI: http://dx.doi.org/10.5772/intechopen.93084*

the parameters according to the exact solution is equal: *a*<sup>2</sup> ¼ 0*:*664; *c* ¼ 1*:*72. Our decision gives *a*<sup>2</sup> ¼ 0*:*6641; *c* ¼ 1*:*7308. Of course, such a good match is due to the fact that these parameters are largely determined by local internal asymptotics, more precisely, derived from the function on the wall. But also within the transition area, the deviation from the exact solution does not exceed 1÷2% (for *φ*<sup>0</sup> and *T*, respectively). Design values of parameters for determining approximations (37) and (38) for *n* = 1 are given in **Table 2**.

If *n* ¼ 0*:*76, this value corresponds to the physical characteristics of the air, and the constant calculation results for the approximation formulas (49) and (50) are shown in **Table 3**.
