**3. Modified method of parameter continuation**

Let us introduce a formal definition of the proposed modified method of parameter continuation (MMPC) for systems of ODEs using the terminology of the perturbation method. It is known [3] that any ODE or system of ODEs may be represented by a normal system of ODEs of the first order in respect to unknown functions <sup>1</sup> *n i i i u u* in the vicinity of regular point in the interval : ]0,1[ :

Applications of 2D Padé Approximants in Nonlinear Shell Theory: Stability Calculation and Experimental Justification 5

$$L u\_i + R\_i \left( \xi, \mu\_1, \dots, \mu\_n \right) + N\_i \left( \xi, \mu\_1, \dots, \mu\_n \right) = g\_i \left( \xi \right), \ L = \frac{d}{d\xi'}, \qquad i = \overline{1, n}, \tag{1}$$

with the BCs on the bounds : 01 

4 Nonlinearity, Bifurcation and Chaos – Theory and Applications

1 2 *m mm Z* , be fixed and let the class

be defined as a class of functions with the properties:

following Montessus de Ballore – type theorem [2]:

.■

**3. Modified method of parameter continuation** 

vicinity of regular point in the interval : ]0,1[

deg*Q m <sup>m</sup>* , i.e. 1 1 22 deg ,0 , deg 0, *Qz m Q z m m m* ;

1 2 *Pz z*, is an entire function;

, : , , *m m*

functions 1 2 *Pz P z* ,0 , 0, and polynomials 1 2 ,0 , 0, *Qz Q z m m* are not equal to zero

The most important theorem for using 2-D PAs for meromorphic continuation is the

1. For all 1 2 *n nn* ' min , that are large enough, there is a unique Padé approximant

2. The sequence *nm F* for 1 2 *n nn* ' min , converges uniformly to function 1 2 *Fz z*,

inside the compact subsets of <sup>2</sup> \ 0 *GC Qm* . For any compact <sup>2</sup> *E C* the following

lim 0,

*<sup>n</sup> m n <sup>E</sup>*

*Q q*

*<sup>n</sup> nm <sup>E</sup>*

*F F*

lim 0,

This is an analog of the classical Montessus de Ballore theorem for the convergence of the

Let us introduce a formal definition of the proposed modified method of parameter continuation (MMPC) for systems of ODEs using the terminology of the perturbation method. It is known [3] that any ODE or system of ODEs may be represented by a normal

system of ODEs of the first order in respect to unknown functions <sup>1</sup>

:

1/ '

*n*

1/ '

*n*

*M M C F Fz z*

**Theorem.** Let 1 2 , *<sup>m</sup> Fz z M* be given by the power series, <sup>2</sup>

*nm n n F Pq* for each of the determinative sets , , 1,2 *<sup>j</sup> I nm j* ;

'

'

2 1 2 1 2

*m*

*Pz z*

*Q zz*

1 2

1 2 *m mm Z* , be fixed and

*n*

*i i i u u*

 in the

Let <sup>2</sup>

*Qm* 0,0 1 ;

 <sup>2</sup> 1 2 *n nn Z* , . Then:

relationships are true:

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

rows of Padé tables.

simultaneously.

$$\left.G\_{j}(\mu\_{1},...,\mu\_{n})\right|\_{\mathfrak{X}\Omega} = 0,\ j = \overline{1,n} \tag{2}$$

Here *L* and *Ri* are the linear differential operators, whereas *Ni* and *Gj* are the non-linear differential operators. We assume also that point 0 0 belongs to closure , and *Ri* , *Ni* and *Gj* are the holomorphic functions for <sup>1</sup> *n <sup>i</sup> <sup>i</sup> <sup>u</sup>* .

Considering <sup>1</sup> *n i i i u u* and their derivatives as independent arguments, we introduce operators *Ri* , *Ni* , *F* and *Gj* as the multidimensional Taylor series

$$R\_i + N\_i = \sum\_{j=1}^{n} \left( N\_{ij} u\_j + \frac{1}{2!} \sum\_{p=1}^{n} N\_{ijp} u\_j u\_p + \dots \right), \quad i = \overline{1, n}\_{\prime} \tag{3}$$

$$F = \left(F\_0 L u\_1 + \frac{1}{2!} \sum\_{p=1}^n F\_{0p} u\_p L u\_1 + \dots \right) + \sum\_{j=1}^n \left(F\_j u\_j + \frac{1}{2!} \sum\_{p=1}^n F\_{jp} u\_j u\_p + \dots \right),$$

$$G\_j = \sum\_{q=1}^n \left(G\_{jq} \left(u\_q - u\_q \Big|\_{\partial \Omega} \right) + \frac{1}{2!} \sum\_{p=1}^n G\_{jqp} \left(u\_q - u\_q \Big|\_{\partial \Omega} \right) \left(u\_p - u\_p \Big|\_{\partial \Omega} \right) + \dots \right) j = \overline{1, n}. \tag{4}$$

We also introduce the following power series

$$N\_{\vec{\eta}} = \sum\_{r=0}^{\alpha} N\_{\vec{\eta}}^{r} \xi^{r} \ \ N\_{\vec{\eta}\vec{p}} = \sum\_{r=0}^{\alpha} N\_{\vec{\eta}\vec{p}}^{r} \xi^{r} \ \dots \ F\_{\vec{\eta}} = \sum\_{r=0}^{\alpha} F\_{\vec{\eta}}^{r} \xi^{r} \ \ F\_{\vec{\eta}\vec{p}} = \sum\_{r=0}^{\alpha} F\_{\vec{\eta}\vec{p}}^{r} \xi^{r} \ \dots \ g\_{i} = \sum\_{j=0}^{\alpha} g\_{i\vec{\eta}} \xi^{i} \ \ i\_{\ast} j \ \eta = \overline{1 \,\wedge} \ \eta \tag{5}$$

To implement the MMPC, we introduce parameter as follows

$$
\mu\_i = \sum\_{j=0}^{\infty} \mu\_{ij}^M \mathfrak{a}^j,\tag{6}
$$

$$\mathcal{L}\boldsymbol{u}\_{i} = \mathcal{E}\left(\mathcal{g}\_{i} - \mathcal{R}\_{i}\left(\boldsymbol{u}\_{1}, \dots, \boldsymbol{u}\_{n}\right) - \mathcal{N}\_{i}\left(\boldsymbol{u}\_{1}, \dots, \boldsymbol{u}\_{n}\right)\right), \quad i = \overline{1, n}, \tag{7}$$

$$\mathcal{G}\_{j}(\boldsymbol{u}\_{1}\big|\_{\partial\Omega}, \dots, \boldsymbol{u}\_{n}\big|\_{\partial\Omega})\Big|\_{\partial\Omega} = 0, \; j = \overline{1, n}$$

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

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

0 <sup>0</sup> 10 0 <sup>0</sup> : 0, ( ,..., ) 0 , 1, , *M MM M i i n ii Lu G u u uu in* 1 1 0 0 0 1 1 1 : ... 0, 2! *n n <sup>M</sup> <sup>M</sup> M M i i ir r irp r p r p Lu g N u N u u* <sup>1</sup> <sup>0</sup> *<sup>H</sup> i u* 1 1 01 1 <sup>1</sup> ... , <sup>1</sup> 2! *<sup>j</sup> n n <sup>M</sup> j j i ij ir r irp r p jr p u gN u N u u j i n* 1, , <sup>2</sup> 2 1 10 01 1 1 1 : ... 0, 2! *n n <sup>M</sup> <sup>M</sup> MM MM i ir r irp r p r p r p Lu N u N u u u u* <sup>2</sup> <sup>0</sup> *<sup>M</sup> i u* (8) 2 2 01 0 1 1 <sup>1</sup> ... 1 2 2! *<sup>n</sup> s j n n M s j j i ir rj rl l rlq l q sr j l q u N g Nu N u u j sj* 2 1 0 1 1 1 1 2! 1 2 2! *<sup>n</sup> s j n n <sup>s</sup> j j irp p rj rl l rlq l q p j l q N u g Nu N u u j sj* 2 0 11 <sup>1</sup> ... ..., 1, . 1 2 2! *s j n n j j r pj pl l plq l q j lq u g N u N u u i n j sj* 

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

0 0 0 1 1

*n n <sup>p</sup> r pl l plq l q l q <sup>g</sup> u N u N <sup>u</sup> <sup>u</sup>* 

variable.

equation (9) becomes

*g*

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

Analysis of the obtained approximation suggests that it gives the exact value of coefficients in the power of the independent variable to the extent equal to the order of approximation (taking into account the expansion in power series of expressions in the equation). This guarantees stability of the computation with a limit-order approximation of the independent

One of the possible fields for application of the proposed approach are the nonlinear problems of plates and shells theory. The equations of statics of the geometrically nonlinear thin-walled structures can be reduced to the resolving equations which contain the products and squares of the desired functions and their derivatives [4]. In this case the solution of

1 1

 

 

*n n*

1 2!

*ir r irp r p*

(10)

*i n* 1, .

or 2-D PAs. 2-D

01 0 0

2 11 1

1 1 2 2 2! *n n <sup>i</sup>*

000 0

1 1 2 2 2! *nn n <sup>r</sup>*

11 1

*rl q*

1 11 2! 2 2 2! *n nn <sup>r</sup>*

> 0 0 0 1 1

*u Nu N u u* 

The approximation thus obtained is converted to 1-D PAs in respect to

*n n <sup>p</sup> r pl l plq l q l q*

*i i i ir r irp r p r p*

1 1

*ir rl l rlq l q*

*<sup>g</sup> <sup>N</sup> Nu N u u* 

0 00 0

*irp p rl l rlq l q*

*<sup>g</sup> N u Nu N u u* 

1 1 ..., 2 2 2!

PAs in the form proposed by V. Vavilov [2] is very promising for the use as an analytical continuation. This technique allows us to choose the coefficients of 2-D Taylor series for the construction of unambiguous 2-D PAs with a given structure of the numerator and denominator. It also ensures optimal PAs features in the sense of the theorem of Montessus de Ballore-type. This means homogenous convergence of PAs to the approximated function with an increase of the degree of the numerator and denominator in all points of its meromorphic area. It should be noted that direct application of 2-D PAs does not lead to the anticipated merging of 1-D approximations. This is due to the initial requirements of the 2-D

1 11

*p lq*

*r p*

*Nu N u u*

*u u g Nu N u u*

0

 

*g*

*i n* 1, .

Stability Calculation and Experimental Justification 7

Summation in (8) of the coefficients with the same degrees of for 1 gives

 <sup>0</sup> 0 0 ... *i i u u* 1 00 0 1 1 <sup>1</sup> <sup>0</sup> ... 0 ... 2! *n n i ir r irpr p r p g Nu N u u* 2 11 1 1 1 1 1 <sup>0</sup> ... 2 2 2! *n n <sup>i</sup> ir r irp r p r p g Nu N u u* 00 0 0 11 1 1 1 ... 2 2 2! *nn n <sup>r</sup> ir rl l rlq l q rl q <sup>g</sup> <sup>N</sup> Nu N u u* (9) 0 00 0 1 11 1 11 2! 2 2 2! *n nn <sup>r</sup> irp p rl l rlq l q p lq <sup>g</sup> N u Nu N u u* 

Applications of 2D Padé Approximants in Nonlinear Shell Theory: Stability Calculation and Experimental Justification 7

$$\left| +\boldsymbol{\mu}\_{r}\Big|\_{\partial\Omega} \left( \frac{\mathcal{S}\_{p0}}{2} - \frac{1}{2} \sum\_{l=1}^{n} \Big( \left. \boldsymbol{N}\_{pl}^{0} \boldsymbol{u}\_{l} \right|\_{\partial\Omega} + \frac{1}{2!} \sum\_{q=1}^{n} \left. \boldsymbol{N}\_{plq}^{0} \boldsymbol{u}\_{l} \Big|\_{\partial\Omega} \boldsymbol{u}\_{q} \Big|\_{\partial\Omega} \right) \right) \Big|\_{\partial\Omega} \right) + \dots \Big| + \dots \Big| + \dots \\ \left. \left. + \overline{\boldsymbol{1}\_{i} \boldsymbol{n}\_{i}} \right]$$

6 Nonlinearity, Bifurcation and Chaos – Theory and Applications

 1

*j sj* 

2

0

*j* 

<sup>0</sup> 10 0 <sup>0</sup> : 0, ( ,..., ) 0 , 1, , *M MM M i i n ii*

 *Lu G u u uu in* 

> 

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

<sup>2</sup> <sup>0</sup> *<sup>M</sup>*

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

*ir r irp r p*

(9)

1 1 ... 2 2 2!

<sup>1</sup> <sup>0</sup> *<sup>H</sup>*

*i u*

*i u*

 

 for 

1 gives

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

(8)

1 0 0 0 1 1

 *Lu g N u N u u* 

*<sup>j</sup> n n <sup>M</sup> j j*

01 1

 <sup>2</sup> 2 1 10 01

2

0 11

1 1

2 11 1

11 1

1 11 2! 2 2 2! *n nn <sup>r</sup>*

*nn n <sup>r</sup>*

*rl q*

*r p*

*g Nu N u u*

*i ir r irpr p*

*n n <sup>i</sup>*

*Nu N u u*

00 0 0

1 11

*p lq*

*n n*

Summation in (8) of the coefficients with the same degrees of

1 00

*g*

*<sup>n</sup> s j n n M s j j*

2

1 1 2! 1 2 2!

1

 *Lu N u N u u u u* 

1 1

*r p*

*i ij ir r irp r p jr p u gN u N u u*

1 : ... 0, 2! *n n <sup>M</sup> <sup>M</sup> M M i i ir r irp r p r p*

: ... 0, 2! *n n <sup>M</sup> <sup>M</sup> MM MM i ir r irp r p r p*

01 0 1 1

1 0 1 1

*p j l q*

*s j n n j j r pj pl l plq l q j lq*

*<sup>n</sup> s j n n <sup>s</sup> j j*

*i ir rj rl l rlq l q sr j l q u N g Nu N u u j sj*

*irp p rj rl l rlq l q*

*u g N u N u u i n*

<sup>0</sup> 0 0 ... *i i u u*

<sup>1</sup> <sup>0</sup> ... 0 ... 2!

 

 

> 1 1 1 1 <sup>0</sup> ... 2 2 2!

*ir rl l rlq l q*

*<sup>g</sup> <sup>N</sup> Nu N u u* 

0 00 0

*irp p rl l rlq l q*

*<sup>g</sup> N u Nu N u u* 

*r p*

 

*N u g Nu N u u j sj*

 *i n* 1, ,

0

1

1

2

Analysis of the obtained approximation suggests that it gives the exact value of coefficients in the power of the independent variable to the extent equal to the order of approximation (taking into account the expansion in power series of expressions in the equation). This guarantees stability of the computation with a limit-order approximation of the independent variable.

One of the possible fields for application of the proposed approach are the nonlinear problems of plates and shells theory. The equations of statics of the geometrically nonlinear thin-walled structures can be reduced to the resolving equations which contain the products and squares of the desired functions and their derivatives [4]. In this case the solution of equation (9) becomes

01 0 0 0 1 1 1 2! *n n i i i ir r irp r p r p u u g Nu N u u* 2 11 1 1 1 1 1 2 2 2! *n n <sup>i</sup> ir r irp r p r p g Nu N u u* 000 0 11 1 1 1 2 2 2! *nn n <sup>r</sup> ir rl l rlq l q rl q <sup>g</sup> <sup>N</sup> Nu N u u* (10) 0 00 0 1 11 1 11 2! 2 2 2! *n nn <sup>r</sup> irp p rl l rlq l q p lq <sup>g</sup> N u Nu N u u* 0 0 0 1 1 1 1 ..., 2 2 2! *n n <sup>p</sup> r pl l plq l q l q g u Nu N u u i n* 1, .

The approximation thus obtained is converted to 1-D PAs in respect to or 2-D PAs. 2-D PAs in the form proposed by V. Vavilov [2] is very promising for the use as an analytical continuation. This technique allows us to choose the coefficients of 2-D Taylor series for the construction of unambiguous 2-D PAs with a given structure of the numerator and denominator. It also ensures optimal PAs features in the sense of the theorem of Montessus de Ballore-type. This means homogenous convergence of PAs to the approximated function with an increase of the degree of the numerator and denominator in all points of its meromorphic area. It should be noted that direct application of 2-D PAs does not lead to the anticipated merging of 1-D approximations. This is due to the initial requirements of the 2-D approximation to ensure its transition to 1-D in the case when the second variable is equal to zero [1]. At the same time it is necessary to ensure such a transition, when the parameter is equal to one. This can be achieved by combining this method with 2-D PAs from a converted parameter which maps the unit to zero.

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

(11)

is the factor at the highest derivative, as shown in Fig. 1

= 0.1 and approximate solutions (1 – three terms

(13)

(14)

(12)

' 1, (0) 0, 0 1, 0. *z z z x* 

2 <sup>1</sup> <sup>1</sup> 1 exp ... ( 1) ... . 2 ! *<sup>n</sup> x xx <sup>n</sup> <sup>x</sup>*

account many terms in (2) to obtain an acceptable and reliable approximation. Thus, the

1 <sup>1</sup> ' , *<sup>z</sup>*

1 0

. *i i i z z* 

for ADM, 3 – three terms HAM, 4 – *x z* for HAM, 5 – 2-D PAs for MMPC, ADM and

 

*n*

accuracy of the used truncated Taylor series is not uniform according to *x* value.

Eq. (12) shows that the exact solution is regular for all real positive *x* for

general term of power series (12) grows rapidly when *x*

= 0.1. The exact solution of this BVP follows

*z*

**Figure 1.** The exact solution (solid line) of Eq. (1) for

as follows

*z*

ADM, 2 – <sup>1</sup> *z*

and suppose

HAM).

Let us introduce parameter 1

where natural small parameter

for  Stability Calculation and Experimental Justification 9

, and we have to take into

0 . But the
