**4. Application of Padé approximants**

#### **4.1 Using of TPPAs in boundary-value problems**

For boundary-value problems, we assume that there exist two asymptotics for limit values of the parameter. In this case, the method of matching of asymptotic expansions is usually used [4]. However, for correct application of the matching method, it is necessary to know the matching point or, at least, the domain of overlapping of asymptotics. An exact description of the transition layer 0 <ε< ∞ exists only in the cases where solutions with different behaviors on opposite sides of the layer can be matched by a special function (e.g., the Airy function).

For the matching of nonoverlapping asymptotics, a method based on TPPAs has recently been developed. In [15, 21, 23], this method was applied for the construction of thermal profiles in a boundary layer of gas. In [2, 6], this method allowed one to examine the heat exchange in hypersonic boundary layers.

Two-point Padé approximations (TPPAs) are defined in Section 3.2 [see formulas (2)–(4)]. As an example of application of TPPAs, we consider the Airy boundary-value problem [4, 10, 31]:

$$
\lambda y'' - \lambda^2 \mathbf{x} y = \mathbf{g}(\mathbf{x}) y \text{ as } \lambda \to \infty \tag{13}
$$

with boundary conditions

$$y(\mathbf{0}) = \mathbf{1}, \ y(\boldsymbol{\omega}) = \mathbf{0} \tag{14}$$

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

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

$$U'' - sU = 0, \ U(0) = 1, \ U(\ast \omega) = 0 \tag{15}$$

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

$$y(\mathbf{x}) = U(\mathbf{s}) \left[ \mathbf{1} + \mathcal{O}\left(-\lambda^{-1}\right) \right] \text{ as } \mathbf{s} = \mathbf{x}\lambda^{2/3}. \tag{16}$$

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

$$U^i = \mathbf{1} - a\mathbf{s} + \frac{\mathbf{1}}{6}\mathbf{s}^3 + \mathcal{O}(\mathbf{s}^4) \tag{17}$$

The exterior asymptotic has the form of an exponential function:

$$U^{\epsilon} = b s^{-\aleph\_4} \exp\left(-\frac{2}{3} s^{\flat\_2}\right) \left[1 - \frac{5}{48} s^{-\aleph\_2} + O(s^{-3})\right] \tag{18}$$

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

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 defini-

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

*Mathematical Theorems - Boundary Value Problems and Approximations*

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

*k* coefficients which are determined from the condition

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

TPPA is a rational function of the form:

**4. Application of Padé approximants**

boundary-value problem [4, 10, 31]:

with boundary conditions

function).

**82**

**4.1 Using of TPPAs in boundary-value problems**

*i*¼0

*i*¼0

*<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>*

<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><sup>n</sup>*þ*m*þ<sup>1</sup> � �

For boundary-value problems, we assume that there exist two asymptotics

For the matching of nonoverlapping asymptotics, a method based on TPPAs has recently been developed. In [15, 21, 23], this method was applied for the construction of thermal profiles in a boundary layer of gas. In [2, 6], this method allowed

Two-point Padé approximations (TPPAs) are defined in Section 3.2 [see formulas (2)–(4)]. As an example of application of TPPAs, we consider the Airy

for limit values of the parameter. In this case, the method of matching of asymptotic expansions is usually used [4]. However, for correct application of the matching method, it is necessary to know the matching point or, at least, the domain of overlapping of asymptotics. An exact description of the transition layer 0 <ε< ∞ exists only in the cases where solutions with different behaviors on opposite sides of the layer can be matched by a special function (e.g., the Airy

one to examine the heat exchange in hypersonic boundary layers.

*<sup>y</sup>*<sup>00</sup> � *<sup>λ</sup>*<sup>2</sup>

*ciε<sup>i</sup>* at *<sup>ε</sup>* ! 0, (9)

*diε*�*<sup>i</sup>* at *<sup>ε</sup>* ! <sup>∞</sup> (10)

<sup>1</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>ε</sup>* <sup>þ</sup> … <sup>þ</sup> *bmε<sup>m</sup>* , (11)

*xy* ¼ *g x*ð Þ*y* as *λ* ! ∞ (13)

*y*ð Þ¼ 0 1, *y*ð Þ¼ ∞ 0 (14)

(12)

tion following [25]. Let

The transition layer is defined by the domain, where *x* ¼ *O λ* �2*=*3 � � Airy function approaches with TPPA:

$$U\_{\rm at} = \frac{\mathbf{1} - a\mathbf{s} + \frac{2}{3}s^{\diamondsuit} - \frac{2}{3}as^{\diamondsuit} + \frac{32}{5}as^{\diamondsuit}}{\mathbf{1} + \frac{32}{5}\frac{a}{b}s^{\diamondsuit}} \exp\left(-\frac{2}{3}s^{\diamondsuit}\right) \tag{19}$$

The TPPA (19) preserves three terms of the asymptotics at both ends and provides accuracy with relative error:

$$\Delta = \frac{|U - U\_a|}{U} \sim \mathbf{1.596}$$

Parameters *a* and *b* are obtained from the integral equations (relations). The relations (20) and (21) can be obtained by multiplying Eq. (18) by 1, *s*, *s* 2, … and then by integrating from 0 to ∞.

$$U'' = sU \Rightarrow \int\_0^\infty U'' ds = \int\_0^\infty sU ds \Rightarrow \int\_0^\infty (U')' ds = \int\_0^\infty sU ds \Rightarrow U'|\_o^\infty = \int\_0^\infty sU ds$$

This is the first integral relation.

$$\int\_0^\infty sU ds = a \tag{20}$$

$$sU'' = s^2 U \Rightarrow \int\_0^\infty sU'' ds = \left| \int s^2 U ds \Rightarrow \begin{vmatrix} s = t, & dt = ds\\ s^2 U ds \Rightarrow \begin{vmatrix} s = t, & dt = ds\\ U'' ds = dV, & V = U' \end{vmatrix} \Rightarrow U'|\_0^\infty - \int\_0^\infty U' ds = 0 \end{vmatrix}$$
 $\frac{1}{s} \int s^2 U ds \Rightarrow \int s^2 U ds = 1$ 

This is the next integral relation.

$$\int\_0^\infty s^2 U ds = 1\tag{21}$$

chosen as the weight, then its inhomogeneity increases the influence of the local

In the illustrated example (5), Eq. (18) TPPA represents a modified (quasifractional) two-point Padé approximant (10) by an exponential weight function, the choice of which is dictated by a kind of exterior asymptotics. Evidently, the TPPAs are not panacea. For example, one of the "bottlenecks" of the TPPAs method is related to the presence of logarithmic components in numerous asymptotic expansions. This problem is the most essential for the TPPAs, because, as a rule, one of the limits *ε* ! 0 *or ε* ! ∞ for a real mechanical problem gives expansions with logarithmic terms or other complicated functions. It is worth noting that in some cases these obstacles may be overcome by using an approximate method of TPPAs' construction by tacking as limit points not *ε* ¼ 0 *and ε* ¼ ∞, but some small and large values. On the other hand, Martin and Baker [32] proposed the so-called quasi-fractional approximants (QAs). Let us suppose that we have a perturbation approach in powers of *ε* for *ε* ! 0 and asymptotic expansions *F*ð Þ*ε* containing, for example, logarithm for *ε* ! ∞. By definition, QA is a ratio R with unknown coefficients *ai*, *bi*, containing both powers of *ε* and *F*ð Þ*ε* . We give this modification of TPPA [2, 14, 15]. Let the series give for Eq. (5). Then the modification of TPPA is

**4.2 Quasi-fractional Padé approximants (modification of TPPA)**

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

region where the inhomogeneity is concentrated.

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

represented by the irrational function:

**mechanics**

where *φ ζ*ð Þ¼ *<sup>ψ</sup><sup>=</sup>* ffiffiffi

a flat plate:

**85**

*F*ð Þ¼ *ε*

P*<sup>m</sup> <sup>k</sup>*¼<sup>0</sup>*akε<sup>k</sup>*

P*<sup>n</sup>*

cation of the method of solving nonlinear algebraic systems [4, 23, 24].

*φ*‴ þ *φφ*<sup>00</sup> ¼ 0;

*φ*ð Þ¼ 0 *φ*<sup>0</sup>

along the flow. The interior asymptotic (ζ ! 0) has the form:

**4.3 Application of TPPAs in problems of incompressible liquid and gas**

Consider the Blasius equation (45), which describes laminar boundary layers on

ð Þ¼ 0 0; *φ*<sup>0</sup>

*<sup>x</sup>* <sup>p</sup> , *<sup>ψ</sup>*ð Þ*<sup>y</sup>* is the stream function, *<sup>ζ</sup>* <sup>¼</sup> *<sup>y</sup>*

*<sup>φ</sup>* <sup>¼</sup> *<sup>a</sup>*2*ζ*<sup>2</sup> � *<sup>a</sup>*<sup>2</sup>

variable, and *x* and *y* are the Cartesian coordinates such that the axis *x* is directed

2

ð Þ¼ ∞ 2

2

ffiffiffiffiffi Re *x* q

<sup>30</sup> *<sup>ζ</sup>*<sup>5</sup> <sup>þ</sup> *<sup>O</sup> <sup>ζ</sup>*<sup>8</sup> � � (26)

*<sup>k</sup>*¼<sup>0</sup>*bkε<sup>k</sup>* exp �

where *k* þ 1 coefficients *ck*,ð Þ *k* ¼ 0, 1, 2, … are determined by means of *l* þ 1 integral equations for function from Eqs. (20) and (21). We notice that exponential terms [multiplier in expressions (17) and (18)] give for *ε* ¼ 0 *and ε* ¼ ∞ coincidence with TPPA (19). When considering the computational aspects of the connection method, it should first be assumed that the system of equations for determining the TPPA parameters is substantially nonlinear. To solve it, we developed a modifi-

X *l*

*ckε<sup>k</sup>* !

, (24)

(25)

is the automodel

*k*¼0

Substituting in Eqs. (20) and (21) instead of *U* (4) interpolation *Ua* (7), calculate using quadrature integration formulas *a* = 0.7287 and *b* = 0.7922.

In the same manner, integral relations with weights *U, U'* can be obtained by part integration. Multiplying Eq. (18) by *U*, *U*<sup>0</sup> , *U*<sup>00</sup> … *,* we get after integration from 0 to ∞,

$$\begin{aligned} UU'' &= sU^2 \Rightarrow \int\_0^\infty UU''ds = \left. \int sU^2ds \Rightarrow \begin{aligned} U &= t, & dt &= dU\\ U''ds &= dV, & V &= U' \end{aligned} \right| \Rightarrow \left. UU'^2 \right|\_0^\infty - \int\_0^\infty U'^2ds\\ &= \left. \int sU^2ds \Rightarrow a - \left. \int\_0^\infty U'^2ds = \left. \int sU^2ds \Rightarrow \int\_0^\infty \left(U^2 + sU^2\right)ds = a \right. \end{aligned} \right| \Rightarrow a = 0$$

This is the first integral relation for the second method of producing it:

$$\begin{aligned} \stackrel{\circ}{\int} \left( U^{\prime 2} + sU^{2} \right) ds &= a \\ U^{\prime}U^{\prime \prime} = sU^{\prime}U \Rightarrow \stackrel{\circ}{\int} \left. U^{\prime}U^{\prime} ds \right| &= \stackrel{\circ}{\int} sU^{\prime}U ds \Rightarrow \left| \begin{aligned} U^{\prime} &= t, & dt &= U^{\prime}ds \\ U^{\prime} &= dJ, & V &= U^{\prime} \end{aligned} \right| \\ \Rightarrow \left. U^{2} \right|\_{0}^{\infty} & - \stackrel{\circ}{\int} U^{\prime}U^{\prime} ds = \stackrel{\circ}{\int} sU^{\prime}U ds \Rightarrow -a^{2} = 2 \stackrel{\circ}{\int} sU^{\prime}U ds \\ \Rightarrow \left| \begin{aligned} s &= t, & dt &= ds \\ U^{\prime}U ds &= dV, & V &= \frac{U^{2}}{2} \end{aligned} \right| \Rightarrow -a^{2} = 2 \left[ \frac{\circ U^{2}}{2} \Big|\_{0}^{\infty} - \stackrel{\circ}{\int} \frac{U^{2}}{2} ds \right] \Rightarrow \int \stackrel{\circ}{\int} U^{2} ds \\ \Rightarrow \left. \right\|\_{0}^{\infty} \end{aligned}$$

And this is the next integral relation for the second method of producing it:

$$\int\_0^\infty U^2 ds = a^2 \tag{23}$$

Using Eq. (19), from Eqs. (22) and (23), we calculate *а* = 0.7277, and *b* = 0.7966.

From the given example, it follows that the features of the asymptotic connection method are the ambiguity of the algorithm, the freedom to choose both the form of TPPAs, integral relations, and methods for calculating the parameters of the TPPAs. The question of choosing integral relations is, in fact, a question of controlling the asymptotic approximation using weights selected to obtain integral relations. Choosing the weight allows you to achieve acceptable accuracy in a particular area of the boundary layer: a weight equal to 1 means that the uniform influence of the entire layer is taken into account; a weight equal to 1, *s*, *s* 2, … increases the influence of the outer region of the layer; and if the desired solution *U*, *U*<sup>0</sup> , *U*<sup>00</sup> is

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

chosen as the weight, then its inhomogeneity increases the influence of the local region where the inhomogeneity is concentrated.

## **4.2 Quasi-fractional Padé approximants (modification of TPPA)**

In the illustrated example (5), Eq. (18) TPPA represents a modified (quasifractional) two-point Padé approximant (10) by an exponential weight function, the choice of which is dictated by a kind of exterior asymptotics. Evidently, the TPPAs are not panacea. For example, one of the "bottlenecks" of the TPPAs method is related to the presence of logarithmic components in numerous asymptotic expansions. This problem is the most essential for the TPPAs, because, as a rule, one of the limits *ε* ! 0 *or ε* ! ∞ for a real mechanical problem gives expansions with logarithmic terms or other complicated functions. It is worth noting that in some cases these obstacles may be overcome by using an approximate method of TPPAs' construction by tacking as limit points not *ε* ¼ 0 *and ε* ¼ ∞, but some small and large values. On the other hand, Martin and Baker [32] proposed the so-called quasi-fractional approximants (QAs). Let us suppose that we have a perturbation approach in powers of *ε* for *ε* ! 0 and asymptotic expansions *F*ð Þ*ε* containing, for example, logarithm for *ε* ! ∞. By definition, QA is a ratio R with unknown coefficients *ai*, *bi*, containing both powers of *ε* and *F*ð Þ*ε* . We give this modification of TPPA [2, 14, 15]. Let the series give for Eq. (5). Then the modification of TPPA is represented by the irrational function:

$$F(\varepsilon) = \frac{\sum\_{k=0}^{m} a\_k \varepsilon^k}{\sum\_{k=0}^{n} b\_k \varepsilon^k} \exp\left(-\sum\_{k=0}^{l} c\_k \varepsilon^k\right),\tag{24}$$

where *k* þ 1 coefficients *ck*,ð Þ *k* ¼ 0, 1, 2, … are determined by means of *l* þ 1 integral equations for function from Eqs. (20) and (21). We notice that exponential terms [multiplier in expressions (17) and (18)] give for *ε* ¼ 0 *and ε* ¼ ∞ coincidence with TPPA (19). When considering the computational aspects of the connection method, it should first be assumed that the system of equations for determining the TPPA parameters is substantially nonlinear. To solve it, we developed a modification of the method of solving nonlinear algebraic systems [4, 23, 24].
