**3. Matching of limiting asymptotic expansions**

### **3.1 Method of asymptotically equivalent functions**

This method was originally proposed by Slepyan and Yakovlev for the inversion of the integral transformations. Here is a description of this method, following [26]. Suppose that the Laplace transform of a function of a real variable *f*(*t*) is

$$F(s) = \bigcap\_{t=0}^{\infty} f(t) \mathbf{e}^{-st} ds.$$

To obtain an approximate expression for the inverse transform, it is necessary to clarify the behavior of the transform to the vicinity of the points *s* = 0 and s = ∞ and to determine whether the nature and location of its singular points are on the exact boundary of the regularity or near it. Then the transform *F*(*s*) is replaced by the function *F*0(*s*), approximated the exact inversion and satisfying the following conditions:

1. Functions *F*0(*s*) and *F*(*s*) are asymptotically equivalent at *s* ! ∞ and *s* ! 0, that is,

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

$$F\_0(s) \sim F(s) \text{ at } s \to 0 \text{ and } s \to \infty.$$

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

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 all real values *s* ≥ 0:

$$\max \left| \frac{F\_0(s, a\_1, a\_2, \dots, a\_k)}{F(s)} - 1 \right| \to \min \tag{8}$$

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

$$\int\_0^\infty F\_0(s)ds = \int\_0^\infty F(s)ds$$

or *F*<sup>0</sup> <sup>0</sup>ðÞ� *s F*<sup>0</sup> ð Þ*s* at *s* ! 0 leads to a rather precise fulfillment of the requirements (8).

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 concepts.

Let's consider the function *f(x).* To describe the ordinal relationships with respect to another function *φ(x),* enter the following definitions:

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

$$f(\mathbf{x}) = O(q(\mathbf{x})) $$

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

$$f(\mathfrak{x}) = o(\mathfrak{q}(\mathfrak{x})) $$

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* .

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

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

$$f(\mathbf{x}) \sim \rho(\mathbf{x}) \text{ if } \frac{f(\mathbf{x})}{\rho(\mathbf{x})} \to \mathbf{1}.$$

Here we use the term "asymptotically equivalent function." Other terms ("reduced method of matched asymptotic expansions" [28], "quasi-fractional approximants" (QAs) [29], and "mimic function" [30]) are also used.

### **3.2 Two-point Padé approximants**

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

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.

*Mathematical Theorems - Boundary Value Problems and Approximations*

**3. Matching of limiting asymptotic expansions**

**3.1 Method of asymptotically equivalent functions**

conditions:

**80**

that is,

This method was originally proposed by Slepyan and Yakovlev for the inversion of the integral transformations. Here is a description of this method, following [26]. Suppose that the Laplace transform of a function of a real variable *f*(*t*) is

*f t*ð Þe�*stds:*

*F s*ðÞ¼

∞ð

0

To obtain an approximate expression for the inverse transform, it is necessary to clarify the behavior of the transform to the vicinity of the points *s* = 0 and s = ∞ and to determine whether the nature and location of its singular points are on the exact boundary of the regularity or near it. Then the transform *F*(*s*) is replaced by the function *F*0(*s*), approximated the exact inversion and satisfying the following

1. Functions *F*0(*s*) and *F*(*s*) are asymptotically equivalent at *s* ! ∞ and *s* ! 0,

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.

### *Mathematical Theorems - Boundary Value Problems and Approximations*

asymptotics and for *ε* ! ∞. The most difficult in the asymptotic approach is the intermediate case of *ε* � 1. In this domain, typically numerical methods work well; however, if the task is to investigate the solution depending on the parameter *ε*, then it is inconvenient to use different solutions in different areas. Construction of a unified solution on the basis of limiting asymptotics is not a trivial task, and for this purpose, one can use a two-point Padé approximants (TPPAs). We give the definition following [25]. Let

$$F(\varepsilon) = \sum\_{i=0}^{\infty} c\_i \varepsilon^i \text{ at } \varepsilon \to 0,\tag{9}$$

This boundary-value problem has the form in terms of Airy function *U*(*s*):

*<sup>y</sup>*ð Þ¼ <sup>x</sup> *<sup>U</sup>*ð Þ<sup>s</sup> <sup>1</sup> <sup>þ</sup> *<sup>O</sup>* �*λ*�<sup>1</sup> � � � � as *<sup>s</sup>* <sup>¼</sup> *<sup>x</sup>λ*2*=*<sup>3</sup>

1 6 *s*

> 48 *s* �3*=*

The asymptotic solution for problems (13) and (14) has the form:

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

The interior asymptotic (*s* ! 0) has the form of a power function:

*<sup>U</sup><sup>i</sup>* <sup>¼</sup> <sup>1</sup> � *as* <sup>þ</sup>

The exterior asymptotic has the form of an exponential function:

3 *s* 1*=*2 � � <sup>1</sup> � <sup>5</sup>

<sup>4</sup> exp � <sup>2</sup>

The transition layer is defined by the domain, where *x* ¼ *O λ*

3 *s* 3*=*<sup>2</sup> � <sup>2</sup> <sup>3</sup> *as*<sup>5</sup>*=*<sup>2</sup> <sup>þ</sup> <sup>32</sup> <sup>5</sup> *as*<sup>4</sup>

<sup>1</sup> <sup>þ</sup> <sup>32</sup> 5 *a b s* 12*=*4

The TPPA (19) preserves three terms of the asymptotics at both ends and

Parameters *a* and *b* are obtained from the integral equations (relations). The

∞ð

*U*<sup>0</sup> ð Þ0 *ds* ¼ ∞ð

0

*s* ¼ *t*, *dt* ¼ *ds U*00*ds* ¼ *dV*, *V* ¼ *U*<sup>0</sup>

*sUds* ) *U*<sup>0</sup>

*sUds* ¼ *a* (20)

� � � � ) *U*<sup>0</sup> j ∞ <sup>0</sup> � ∞ð

j ∞ *<sup>o</sup>* ¼ ð ∞

0

*sUds*

0 *U*0 *ds*

0

*<sup>U</sup>* � <sup>1</sup>*:*5%

<sup>Δ</sup> <sup>¼</sup> j j *<sup>U</sup>* � *Ua*

relations (20) and (21) can be obtained by multiplying Eq. (18) by 1, *s*, *s*

*sUds* )

∞ð

0

� � � �

*<sup>U</sup><sup>e</sup>* <sup>¼</sup> *bs*�<sup>1</sup>

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

Airy function approaches with TPPA:

provides accuracy with relative error:

then by integrating from 0 to ∞.

∞ð

*U*00*ds* ¼

ð ∞

0

∞ð

0 *s* 2 *Uds* )

0

This is the first integral relation.

∞ð

*sU*00*ds* ¼

∞ð

0 *s* 2 *Uds* ¼ 1

0

This is the next integral relation.

*U*<sup>00</sup> ¼ *sU* )

*sU*<sup>00</sup> ¼ *s* 2 *U* )

> ¼ ∞ð

**83**

0 *s* 2 *Uds* )

*Ua* <sup>¼</sup> <sup>1</sup> � *as* <sup>þ</sup> <sup>2</sup>

as *a* ffi 0*:*7290, *b* ffi 0*:*7946.

*=*

*U*<sup>00</sup> � *sU* ¼ 0, *U*ð Þ¼ 0 1, *U*ð Þ¼ ∞ 0 (15)

*:* (16)

<sup>3</sup> <sup>þ</sup> *O s*<sup>4</sup> � � (17)

<sup>2</sup> <sup>þ</sup> *O s*�<sup>3</sup> � � � � (18)

exp � <sup>2</sup> 3 *s* 3*=*2

�2*=*3 � �

� � (19)

2, … and

$$F(\varepsilon) = \sum\_{i=0}^{\infty} d\_i \varepsilon^{-i} \text{ at } \varepsilon \to \infty \tag{10}$$

TPPA is a rational function of the form:

$$f\_{\lfloor n/m \rfloor}(\varepsilon) = \frac{a\_0 + a\_1 \varepsilon + \dots + a\_n \varepsilon^n}{1 + b\_1 \varepsilon + \dots + b\_m \varepsilon^m},\tag{11}$$

*k* coefficients which are determined from the condition

$$(\mathbf{1} + b\_1 \boldsymbol{\varepsilon} + \dots + b\_m \boldsymbol{\varepsilon}^m) \left( c\_0 + c\_1 \boldsymbol{\varepsilon} + c\_2 \boldsymbol{\varepsilon}^2 + \dots \right) = a\_0 + a\_1 \boldsymbol{\varepsilon} + \dots + a\_n \boldsymbol{\varepsilon}^n + \mathbf{O} \left( \boldsymbol{\varepsilon}^{n+m+1} \right) \tag{12}$$

and the remaining coefficients from a similar condition for *ε*�1.
