**2. Padé approximants**

Let us consider Padé approximants (1-D PAs) which allow us to perform somewhat the most natural continuation of the power series. Below, we are going to define the 1-D PAs [1] for a complex variable *z* :

$$F(z) = \sum\_{i=0}^{\infty} f\_i z^i,$$

$$F\_{nm}(z) = \sum\_{i=0}^n p\_i z^i \mid \sum\_{i=0}^m q\_i z^i, \quad q\_0 \equiv 1,$$

where coefficients , *i i p q* are determined from the following condition: the first *m n* 1 components of the expansion of rational function ( ) *nm F z* in the McLaurin series coincide with the same components of *F z*( ) series. Then rational function *nm F* is called [*n/m*] PAs or 1-D [*n/m*] PAs. The set of *nm F* functions for different *m* and *n* forms the so called Padé table.

PAs make a meromorphic continuation of *F* due to the following theorem of Montessus de Ballore [10]:

**Theorem.** Let function *F z*( ) be meromorphic in a closed circle *z r* with *m* different poles *<sup>i</sup> z* of multiplicity *<sup>i</sup>* in this circle

$$0 < \left| z\_1 \right| \le \left| z\_2 \right| \le \dots \le \left| z\_m \right| < r$$

of total multiplicity M, 1 *m i i M* .

Then sequence ( ) *F z NM* converges uniformly to *F z*( ) on compact subsets of this circle without poles, and *<sup>i</sup> z* is attracting zeros of the Padé denominator according to its multiplicity:

$$\lim\_{N \to \infty} F\_{\text{NM}}\left(z\right) = F\left(z\right), \quad \left|z\right| \le r,\\ z \ne z\_{i\text{\textquotedblleft}i} = \overline{1, m}. \tag{1}$$

The most common generalization of PAs are two-dimensional PAs (2-D PAs). For complex variables 1 2 *z z*, let

$$F(z\_1, z\_2) = \sum\_{i=0}^{\infty} f\_{i\vec{j}} z\_1^i z\_2^{\vec{j}}$$

be a holomorphic function near the origin. For any integer sets 1 2 *n nn* , and 1 2 *m mm* , , i.e. for any <sup>2</sup> *nm Z* , , let

Applications of 2D Padé Approximants in Nonlinear Shell Theory: Stability Calculation and Experimental Justification 3

$$R(n,m) = \left\{ r = \frac{p}{q}, p = \sum\_{i=0}^{n\_1} \sum\_{j=0}^{n\_2} p\_{ij} z\_1^i z\_2^j, \quad q = \sum\_{i=0}^{m\_1} \sum\_{j=0}^{m\_2} q\_{ij} z\_1^i z\_2^j, q\_{00} = 1 \right\},$$

be the class of rational functions, i.e. the ratio of 2-D polynomials whose degrees do not exceed 1 2 *n nn* , and 1 2 *m mm* , for each variable. It may be written briefly as deg( ) ,deg( ) *pn qm* .

Each rational function *r Rnm* (, ) may be identified with its power series that converges in some neighborhood of the origin. It should be mentioned that *r pq R* depends on 12 1 2 ( 1)( 1) ( 1)( 1) 1 *nm nn mm* parameters (the coefficients of *p* and *q*).

The set of integer points <sup>2</sup> *I nm Z* , for fixed 1 2 *n nn* , and 1 2 *m mm* , is called the determinative (interpolation) set, if it has the following properties:

$$1. \qquad \dim I(n, m) = \pi\_{nm}.$$

2 Nonlinearity, Bifurcation and Chaos – Theory and Applications

approach.

Ballore [10]:

*<sup>i</sup> z* of multiplicity *<sup>i</sup>*

of total multiplicity M,

variables 1 2 *z z*, let

, i.e. for any <sup>2</sup> *nm Z* , , let

in this circle

1

.

*m i i M*

**2. Padé approximants** 

for a complex variable *z* :

specimens based on holographic interferometry. It is shown that the application of PAs provides sufficient accuracy in the studied area that confirms the advantage of our proposed

Let us consider Padé approximants (1-D PAs) which allow us to perform somewhat the most natural continuation of the power series. Below, we are going to define the 1-D PAs [1]

> 0 () ,*<sup>i</sup> i i F z fz*

0

 

0 0 ( ) / , 1, *n m i i*

where coefficients , *i i p q* are determined from the following condition: the first *m n* 1 components of the expansion of rational function ( ) *nm F z* in the McLaurin series coincide with the same components of *F z*( ) series. Then rational function *nm F* is called [*n/m*] PAs or 1-D [*n/m*] PAs. The set of *nm F* functions for different *m* and *n* forms the so called Padé table. PAs make a meromorphic continuation of *F* due to the following theorem of Montessus de

**Theorem.** Let function *F z*( ) be meromorphic in a closed circle *z r* with *m* different poles

1 2 0 ... *<sup>m</sup> zz zr*

Then sequence ( ) *F z NM* converges uniformly to *F z*( ) on compact subsets of this circle without

lim , , , 1, . *NM <sup>i</sup> <sup>N</sup> F z Fz z rz z i m*

The most common generalization of PAs are two-dimensional PAs (2-D PAs). For complex

1 2 1 2 0 (,) *<sup>i</sup> <sup>j</sup> ij i Fz z f zz* 

 

be a holomorphic function near the origin. For any integer sets 1 2 *n nn* , and 1 2 *m mm* ,

■

poles, and *<sup>i</sup> z* is attracting zeros of the Padé denominator according to its multiplicity:

*nm i i i i F z pz qz q* 

2. 11 22 *n m n m I nm* ,0 , 0, , (this property guarantees that in the case when 1*z* 0 (or 2 *z* 0 ) one would have the classical 1-D rational approximation of Padé type),

$$\mathcal{B}.\qquad n = \left(n\_1, n\_2\right) \in I\left(n, m\right)\_{\prime\prime}$$


$$I\_1(m,m) = \{(i,j):$$

$$
\left\lceil \begin{aligned} 0 \le i \le n\_1, 0 \le j \le n\_2 \end{aligned} \right\rceil \cup \left\lceil \begin{aligned} &n\_1 + 1 \le i \le n\_1 + m\_1, 0 \le j \le m\_2 \end{aligned} \right\rceil \cup \left\lceil \begin{aligned} &i = 0, n\_2 + 1 \le j \le n\_2 + m\_2 \end{aligned} \right\rceil \cup$$

$$I\_2 \begin{pmatrix} n, m \end{pmatrix} = \left\{ (i, j) \,:\, \end{aligned} \right\}$$

$$\left\{ \left[ 0 \le i \le n\_1, 0 \le j \le n\_2 \right] \cup \left[ \left\lceil 0 \le i \le m\_1, n\_2 + 1 \le j \le n\_2 + m\_2 \right\rceil \cup \left[ \left\lceil n\_1 + 1 \le i \le n\_1 + m\_1, j = 0 \right\rceil \right] \right\} \right.$$

The generalized PAs for given 1 2 *n nn* , and 1 2 *m mm* , are defined as the rational function , *nm F Rnm* for which 0 *ij nm TFF* for all *ij I nm* , , , where *ij T* are Taylor's coefficients of the power series for function *F*. The rational function *nm F* is called the 2-D PAs 11 22 *nm nm* , /, of 1 2 *Fz z*, which corresponds to the determinative set *ij I nm* , , .

As in the 1-D case, the existence and uniqueness of PAs (in the sense of the above given definition) for <sup>2</sup> *C* require special type of analysis. It should be mentioned that PAs do not always exist in the sense of the given definition.

Let <sup>2</sup> 1 2 *m mm Z* , be fixed and let the class

$$M\_m = M\_m \left( \mathbb{C}^2 \right) = \left\{ F : F \left( z\_1, z\_2 \right) = \frac{P \left( z\_1, z\_2 \right)}{Q\_m \left( z\_1, z\_2 \right)} \right\}$$

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

<sup>1</sup> ( ,..., ) 0, 1, *Gu u j n j n* (2)

 1 1 , ,..., , ,..., , , 1, , *ii n i n i <sup>d</sup> Lu R u u N u u g L i n <sup>d</sup>*

Here *L* and *Ri* are the linear differential operators, whereas *Ni* and *Gj* are the non-linear

*n <sup>i</sup> <sup>i</sup> <sup>u</sup>* .

*R N N u N uu i n*

 

1 11

*p jp F F Lu F u Lu Fu F uu*

*G Gu u G u u u u j n* 

*N N N N F F F F g g ijp n*

0

*Lu g R u u N u u i n i ii n i n*

<sup>1</sup> ( ,..., ) 0, 1, *Gu u j n j n*

Substituting power series (6) into (7) and splitting it with respect to the powers of

*i ij j u u*

*n nn*

 

1 1 ... ... , 2! 2!

*p p j j jp j p*

(4)

, ,... , ,... , , 1, . *r r r r r r r r <sup>i</sup>*

, *<sup>M</sup> <sup>j</sup>*

(5)

<sup>1</sup> ... , 1, . 2!

as follows

1 1 ,..., ,..., , 1, , (7)

and their derivatives as independent arguments, we introduce

<sup>1</sup> ... , 1, , 2!

(3)

(1)

operators *Ri* , *Ni* , *F* and *Gj* as the multidimensional Taylor series

01 0 1

1 1

 

*q p*

We also introduce the following power series

To implement the MMPC, we introduce parameter

where *Ri* and *Ni* are always the algebraic operators in this case.

*n n*

1 1

*j jq q q jqp q q p p*

0 0 00 0

*ij ij ijp ijp j j jp jp i ij r r rr j*

*n n i i ij j ijp j p j p*

differential operators. We assume also that point 0

and *Gj* are the holomorphic functions for <sup>1</sup>

*n*

*i i i u u*

Considering <sup>1</sup>

with the BCs on the bounds : 01

Stability Calculation and Experimental Justification 5

0 belongs to closure , and *Ri* , *Ni*

 

> , we get

(6)

be defined as a class of functions with the properties:


The most important theorem for using 2-D PAs for meromorphic continuation is the following Montessus de Ballore – type theorem [2]:

**Theorem.** Let 1 2 , *<sup>m</sup> Fz z M* be given by the power series, <sup>2</sup> 1 2 *m mm Z* , be fixed and <sup>2</sup> 1 2 *n nn Z* , . Then:


$$\begin{aligned} \lim\_{n^\circ \to \infty} \left\| \mathbb{Q}\_m - q\_n \right\|\_E^{1/n^\circ} &= \mathbb{O}\_\prime, \\ \lim\_{n^\circ \to \infty} \left\| F - F\_{nm} \right\|\_E^{1/n^\circ} &= \mathbb{O}\_\prime \end{aligned}$$

where *j* = 1, 2 and \* sup \* *<sup>E</sup> z E* .■

This is an analog of the classical Montessus de Ballore theorem for the convergence of the rows of Padé tables.
