**2. Mathematical background: summation of asymptotic series**

### **2.1 Analysis of power series**

We suppose that by the result of the asymptotic study, one obtains the following series:

$$f(\varepsilon) \sim \sum\_{n=0}^{\infty} \mathbb{C}\_n \varepsilon^n \text{ for } \qquad \varepsilon \to 0. \tag{1}$$

whose coefficients are determined from the condition

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

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

Padé coefficients of the numerator and denominator of the PAs.

2. PAs gives meromorphic continuation of a given power series.

PAs is invariant concerning Euler transformation (2).

functions. Let us analyze a function (3). Let

exists, then it is unique.

*q*ð Þ¼ *ε f*

If *c* þ *df*ð Þ 0 6¼ 0, then

provided that there is *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* .

one has the following estimate:

Eq. (7) can be replaced by *f*ð Þ*ε* .

**79**

�1

<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>ε</sup>* <sup>þ</sup> … <sup>þ</sup> *bmε<sup>m</sup>* ð Þ *<sup>c</sup>*<sup>0</sup> <sup>þ</sup> *<sup>c</sup>*1*<sup>ε</sup>* <sup>þ</sup> *<sup>c</sup>*2*ε*<sup>2</sup> <sup>þ</sup> … <sup>¼</sup> *<sup>a</sup>*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>ε</sup>* <sup>þ</sup> … <sup>þ</sup> *anε<sup>n</sup>* <sup>þ</sup> <sup>O</sup> *<sup>ε</sup>n*þ*m*þ<sup>1</sup>

Equating coefficients near the same powers *ε*, one obtains a system of linear algebraic equations. In the case where this system is solvable, one can obtain the

We note some properties of the PAs [5, 13, 19]. If the PAs at the chosen m and n

1. If the PAs sequence converges to some function, the roots of its denominator tend to the poles of the function. This allows for a sufficiently large number of terms to determine the pole and then perform an analytical continuation.

3. PAs of the inverse function is treated as the PAs function inverse itself. This property is called duality and is more exactly formulated as follows. Let

ð Þ*<sup>ε</sup>* and *<sup>f</sup>*ð Þ <sup>0</sup> 6¼ 0, then *<sup>q</sup>*½ � *<sup>n</sup>=<sup>m</sup>* ð Þ¼ *<sup>ε</sup> <sup>f</sup>*

4. Diagonal PAs are invariant under fractional linear transformations of the argument. Suppose that the function is given by their expansion (3). Consider the linear fractional transformation that preserves the origin *<sup>W</sup>* <sup>¼</sup> *<sup>a</sup>ε=*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup><sup>ε</sup>* and the function *<sup>q</sup>*(*W*) *= f*ð Þ*<sup>ε</sup>* . Then *<sup>q</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ¼ *<sup>W</sup> <sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* , provided that one of these approximations exist. In particular, the diagonal

5. Diagonal PAs are invariant under fractional linear transformations of

*<sup>q</sup>*ð Þ¼ *<sup>ε</sup> <sup>a</sup>* <sup>þ</sup> *bf*ð Þ*<sup>ε</sup> <sup>c</sup>* <sup>þ</sup> *df*ð Þ*<sup>ε</sup> :*

*<sup>q</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ¼ *<sup>ε</sup> <sup>a</sup>* <sup>þ</sup> *bf*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>*

6. PAs can get the upper and lower bounds for *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* . For the diagonal PAs,

Typically, this estimate is valid for the function itself, that is, *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* in

7. Diagonal and close to them a sequence of PAs often possesses the property of autocorrection [17, 18]. It consists of the following. To determine the coefficients of the numerator and denominator of PAs, we have to solve

*<sup>c</sup>* <sup>þ</sup> *df*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>*

*<sup>f</sup>*½ � *<sup>n</sup>=n*�<sup>1</sup> ð Þ*<sup>ε</sup>* <sup>≤</sup> *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* <sup>≤</sup> *<sup>f</sup>*½ � *<sup>n</sup>=n*þ<sup>1</sup> ð Þ*<sup>ε</sup> :* (7)

�1

½ � *<sup>n</sup>=<sup>m</sup>* ð Þ*<sup>ε</sup>* (6)

(5)

As is known, the radius of convergence *ε*<sup>0</sup> series (1) is determined by the distance to the nearest singularity of the function *f*(*ε*) on the complex plane. To define *ε*0, the Domb-Sykes plot may be useful [8, 10]. In many cases, one can effectively use the conformal mapping of the series, a fairly complete catalog of which is given in [9]. In particular, it sometimes turns out to be a successful Euler transformation [8, 10], based on the introduction of a new variable:

$$
\tilde{\varepsilon} = \frac{\varepsilon}{1 - \varepsilon/\varepsilon\_0}. \tag{2}
$$

Recast the function *<sup>f</sup>* in terms of <sup>~</sup>*ε*, *<sup>f</sup>* � <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*dn*~*ε<sup>n</sup>*, transfer the singularity at the point ~*ε* ¼ ∞.

A natural generalization of Euler transformation looks as follows:

$$
\tilde{\varepsilon} = \frac{\varepsilon}{\left(\mathbf{1} - \varepsilon/\varepsilon\_0\right)^a},
$$

where *α* is the certain number.

#### **2.2 Padé approximants**

"The coefficients of the Taylor series in the aggregate have a lot more information about the values of features than its partial sums. It is only necessary to be able to retrieve it, and some of the ways to do this is to construct a Padé approximant" [11]. Padé approximants (PAs) allow us to transform of power series to a fractional-rational function. Let us define PAs, following Baker and Graves-Morris [25].

Suppose we are given the power series:

$$f(\varepsilon) = \sum\_{i=1}^{\infty} c\_i \varepsilon^i,\tag{3}$$

PAs can be written as the following expression:

$$f\_{\left[\mathfrak{n}/m\right]}(\varepsilon) = \frac{a\_0 + a\_1\varepsilon + \dots + a\_n\varepsilon^n}{1 + b\_1\varepsilon + \dots + b\_m\varepsilon^m},\tag{4}$$

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

whose coefficients are determined from the condition

approximants for the summations of the asymptotic series. Section 3 discusses the method of combining of internal and external asymptotics (matching method) by means of Padé approximants. In the Section 4, the methods of solving specific problems of mathematical physics and mechanics of fluid and gas are demonstrated.

We suppose that by the result of the asymptotic study, one obtains the following

*Cnε<sup>n</sup>* for *<sup>ε</sup>* ! <sup>0</sup>*:* (1)

*:* (2)

*<sup>n</sup>*¼<sup>0</sup>*dn*~*ε<sup>n</sup>*, transfer the singularity at the

, (3)

<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>ε</sup>* <sup>þ</sup> … <sup>þ</sup> *bmε<sup>m</sup>* , (4)

**2. Mathematical background: summation of asymptotic series**

*<sup>f</sup>*ð Þ� *<sup>ε</sup>* <sup>X</sup><sup>∞</sup>

*Mathematical Theorems - Boundary Value Problems and Approximations*

*n*¼0

transformation [8, 10], based on the introduction of a new variable:

A natural generalization of Euler transformation looks as follows:

<sup>~</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>*

"The coefficients of the Taylor series in the aggregate have a lot more information about the values of features than its partial sums. It is only necessary to

be able to retrieve it, and some of the ways to do this is to construct a Padé

to a fractional-rational function. Let us define PAs, following Baker and

approximant" [11]. Padé approximants (PAs) allow us to transform of power series

*<sup>f</sup>*ð Þ¼ *<sup>ε</sup>* <sup>X</sup><sup>∞</sup>

*i*¼1

*<sup>f</sup>*½ � *<sup>n</sup>=<sup>m</sup>* ð Þ¼ *<sup>ε</sup> <sup>a</sup>*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>ε</sup>* <sup>þ</sup> … <sup>þ</sup> *anε<sup>n</sup>*

*ciε<sup>i</sup>*

ð Þ <sup>1</sup> � *<sup>ε</sup>=ε*<sup>0</sup> *<sup>α</sup>* ,

Recast the function *<sup>f</sup>* in terms of <sup>~</sup>*ε*, *<sup>f</sup>* � <sup>P</sup><sup>∞</sup>

where *α* is the certain number.

Suppose we are given the power series:

PAs can be written as the following expression:

**2.2 Padé approximants**

Graves-Morris [25].

**78**

As is known, the radius of convergence *ε*<sup>0</sup> series (1) is determined by the distance to the nearest singularity of the function *f*(*ε*) on the complex plane. To define *ε*0, the Domb-Sykes plot may be useful [8, 10]. In many cases, one can effectively use the conformal mapping of the series, a fairly complete catalog of which is given in [9]. In particular, it sometimes turns out to be a successful Euler

> <sup>~</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>* 1 � *ε=ε*<sup>0</sup>

Section 5 presents a discussion of the obtained.

**2.1 Analysis of power series**

series:

point ~*ε* ¼ ∞.

$$(\mathbf{1} + b\_1 \boldsymbol{\varepsilon} + \dots + b\_m \boldsymbol{\varepsilon}^m) \left( c\_0 + c\_1 \boldsymbol{\varepsilon} + c\_2 \boldsymbol{\varepsilon}^2 + \dots \right) = a\_0 + a\_1 \boldsymbol{\varepsilon} + \dots + a\_n \boldsymbol{\varepsilon}^n + \mathcal{O} \left( \boldsymbol{\varepsilon}^{n+m+1} \right) \tag{5}$$

Equating coefficients near the same powers *ε*, one obtains a system of linear algebraic equations. In the case where this system is solvable, one can obtain the Padé coefficients of the numerator and denominator of the PAs.

We note some properties of the PAs [5, 13, 19]. If the PAs at the chosen m and n exists, then it is unique.


$$q(\varepsilon) = f^{-1}(\varepsilon) \text{ and} \\ f(\mathbf{0}) \neq \mathbf{0}, \text{then} \\ q\_{[n/m]}(\varepsilon) = f^{-1}\_{[n/m]}(\varepsilon) \tag{6}$$


$$q(\varepsilon) = \frac{a + bf(\varepsilon)}{c + df(\varepsilon)} \cdot \varepsilon$$

If *c* þ *df*ð Þ 0 6¼ 0, then

$$q\_{\left[n/n\right]}(\varepsilon) = \frac{a + b f\_{\left[n/n\right]}(\varepsilon)}{c + d f\_{\left[n/n\right]}(\varepsilon)}.$$

provided that there is *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* .

6. PAs can get the upper and lower bounds for *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* . For the diagonal PAs, one has the following estimate:

$$f\_{\left[n/n-1\right]}(\varepsilon) \le f\_{\left[n/n\right]}(\varepsilon) \le f\_{\left[n/n+1\right]}(\varepsilon). \tag{7}$$

Typically, this estimate is valid for the function itself, that is, *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* in Eq. (7) can be replaced by *f*ð Þ*ε* .

7. Diagonal and close to them a sequence of PAs often possesses the property of autocorrection [17, 18]. It consists of the following. To determine the coefficients of the numerator and denominator of PAs, we have to solve

systems of linear algebraic equations. This is an ill-posed procedure, so the coefficients of PAs can be determined with large errors. However, these errors in a certain sense are of self-consistent, so the accuracy of PAs is high. This is the radical difference the PAs from the Taylor series, the calculation error of which only increases with increasing number of terms.

*F*0ðÞ� *s F s*ð Þ at *s* ! 0 and *s* ! ∞*:*

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

the regularity, coincide.

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

max

� � � �

∞ð

0

respect to another function *φ(x),* enter the following definitions:

if ∀*δ*>0∃*A* : j j *x* � *x*<sup>0</sup> <*δ* ) j j *f x*ð Þ ≤ *A*j j *φ*ð Þ *x* .

if ∀*δ*>0∃*ε* : j j *x* � *x*<sup>0</sup> < *δ* ) j j *f x*ð Þ ≤ *ε φ*j j ð Þ *x* .

**3.2 Two-point Padé approximants**

Here *A* is a finite number, and *ε*, *δ* are infinitely small.

all real values *s* ≥ 0:

or *F*<sup>0</sup>

concepts.

that is,

**81**

implementation of equalities

<sup>0</sup>ðÞ� *s F*<sup>0</sup>

requirements (8).

2. Singular points of functions *F*0(*s*) and *F*(*s*), located on the exact boundary of

The free parameters of the function *F*0(*s*) are chosen so as to satisfy the conditions of the good approximation of *F*(*s*) in the sense of minimum relative error for

*F s*ð Þ � <sup>1</sup>

∞ð

*F s*ð Þ*ds*

0

ð Þ*s* at *s* ! 0 leads to a rather precise fulfillment of the

Constructed in such a way function *F*0*(s)* is called asymptotically equivalent function for *F(s)* (AEF). Let's dwell on the terminology. In the following sections, we will use the symbols of ordinal relations. We will give strict definitions of these

Let's consider the function *f(x).* To describe the ordinal relationships with

**Definition 1.** Let us say that *f(x)* is a value of order *φ(x)* at *x* ! *x*0, that is,

*f x*ð Þ¼ *O*ð Þ *φ*ð Þ *x*

*f x*ð Þ¼ *o*ð Þ *φ*ð Þ *x*

**Definition 3.** Let us say that *f (x)* is asymptotically equal to *φ(x)* at *x* ! *x*0,

*<sup>φ</sup>*ð Þ *<sup>x</sup>* ! <sup>1</sup>*:*

*f x*ð Þ� *<sup>φ</sup>*ð Þ *<sup>x</sup>* if *f x*ð Þ

Here we use the term "asymptotically equivalent function." Other terms ("reduced method of matched asymptotic expansions" [28], "quasi-fractional

The analysis of numerous examples confirms "complementarity principle": if for *ε* ! 0, one can construct a physically meaningful asymptotics, there is a nontrivial

approximants" (QAs) [29], and "mimic function" [30]) are also used.

**Definition 2.** Let us say that *f(x)* is a value of order less than *φ(x)* at *x* ! *x*0, that is,

� � �

� ! min (8)

*F*0ð Þ *s*, *α*1, *α*2, … , *α<sup>k</sup>*

Condition (8) is achieved by variation of free parameters *αk*. Often the

*F*0ð Þ*s ds* ¼

Autocorrection property is verified for a number of special functions. At the same time, even for elliptic functions, the so-called Froissart doublets phenomenon arises [26]. Thus, in general, having no information about the location of the poles of the PAs, but relying solely on the very PAs (computed exactly as you wish), we cannot say that you have found a good approximated function. Now consider the question: In what sense the available mathematical results on the convergence of the PAs can facilitate the solution of practical problems? Gonchar's theorem [16] states: If none of the diagonal PAs *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* has poles in the circle of radius *<sup>R</sup>*, then the sequence *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* is uniformly convergent in the circle to the original function *<sup>f</sup>*ð Þ*<sup>ε</sup>* . Moreover, the absence of poles of the sequence of the *<sup>f</sup>*½ � *<sup>n</sup>=<sup>n</sup>* ð Þ*<sup>ε</sup>* in a circle of radius *<sup>R</sup>* confirms convergence of the Taylor series in the circle. Since the diagonal PAs is invariant under fractional linear maps *ε* ! *ε=*ð Þ *aε* þ *b* , the theorem is true for any open circle containing the point of decomposition, and for any area, which is the union of these circles. A significant drawback in practice is the need to check all diagonal PAs. The fact is that if a circle of radius R has no poles only for a subsequence of the sequence of diagonal PAs, then the uniform convergence to its original holomorphic in the disk is guaranteed only with *r*<*r*0, where 0*:*583<*r*<sup>0</sup> <0*:*584 [27]. How can we use these results? Suppose that there are a few terms of the perturbation series and one wants to estimate its radius of convergence *R*. Consider the interval [0,ε0], where the truncated perturbation series and the diagonal PAs of the maximal possible order differ by no more than 5% (adopted in the engineering accuracy of the calculations). If none of the previous diagonal PAs does not have in a circle of radius *ε*<sup>0</sup> poles, then it is a high level of confidence to assert that *R*≥*ε*0.
