**4. Stability investigation**

For the analytic continuation of meromorphic solutions to the region and acceleration of the convergence we use 2-D PAs. To do this, the resulting series are reconstructed in rational functions of the form

$$\mu\_i = \sum\_{j=0}^{m\_1} \sum\_{k=0}^{n\_1} P\_{ijk} \left( P \right) \varepsilon^j \xi^k \bigg/ \sum\_{j=0}^{m\_2} \sum\_{k=0}^{n\_2} Q\_{ijk} \left( P \right) \varepsilon^j \xi^k \,, Q\_{i00} \equiv 1 \,,$$

where *P* is the parameter of loading.

Since the proposed method is modified, all functions belonging to the boundary value problems can be expanded in powers of the independent variables and parameters. For 1 we obtain

$$\mu\_i = \sum\_{j=0}^{m\_3} \sum\_{k=0}^{n\_1} \overline{P}\_{ijk} P^j \xi^k \bigg/ \sum\_{j=0}^{m\_4} \sum\_{k=0}^{n\_2} \overline{Q}\_{ijk} P^j \xi^k \cdot \eta$$

Rational functions have singular points which are determined by equating of the denominator to zero. According to the theory of bifurcation of solutions of ordinary differential equations, at these points either bifurcations or limit states are achieved. So we can get the estimation of critical point localization by solving equations of the form

$$\min\_{i,\xi} P: \sum\_{j=0}^{m\_4} \sum\_{k=0}^{m\_2} \overline{Q}\_{ijk} P^j \xi^k = 0 \ .$$

In practice this equation is transformed to its counterpart simpler form with respect to its characteristic point 0 (which usually corresponds to the point of maximum transverse displacement for thin shells)

$$\sum\_{j=0}^{m\_4} \sum\_{k=0}^{n\_2} \overline{Q}\_{ijk} P^j \xi\_0^k = 0 \ . $$

### **Example**

Let us consider the computational aspects of the proposed approach. We consider three types of PAs with respect to the independent variable, on the specified parameters, and 2-D.

A typical behavior of the approximations for the BVP is governed by the following problem

Applications of 2D Padé Approximants in Nonlinear Shell Theory: Stability Calculation and Experimental Justification 9

$$\begin{aligned} \varepsilon z' + z &= 1, \\ z(0) = 0, \quad 0 < \varepsilon &< 1, \quad \chi \ge 0. \end{aligned} \tag{11}$$

where natural small parameter is the factor at the highest derivative, as shown in Fig. 1 for = 0.1. The exact solution of this BVP follows

$$z = 1 - \exp\left(-\frac{\chi}{\varepsilon}\right) = \frac{\chi}{\varepsilon} - \frac{\chi^2}{2\varepsilon} + \dots + (-1)^{n+1} \frac{1}{n!} \left(\frac{\chi}{\varepsilon}\right)^n + \dots \tag{12}$$

Eq. (12) shows that the exact solution is regular for all real positive *x* for 0 . But the general term of power series (12) grows rapidly when *x* , and we have to take into account many terms in (2) to obtain an acceptable and reliable approximation. Thus, the accuracy of the used truncated Taylor series is not uniform according to *x* value.

**Figure 1.** The exact solution (solid line) of Eq. (1) for = 0.1 and approximate solutions (1 – three terms ADM, 2 – <sup>1</sup> *z* for ADM, 3 – three terms HAM, 4 – *x z* for HAM, 5 – 2-D PAs for MMPC, ADM and HAM).

Let us introduce parameter 1 as follows

$$z^\* = \frac{1}{\varepsilon} - \varepsilon\_1 \frac{z}{\varepsilon'} \tag{13}$$

and suppose

8 Nonlinearity, Bifurcation and Chaos – Theory and Applications

converted parameter which maps the unit to zero.

**4. Stability investigation** 

where *P* is the parameter of loading.

functions of the form

1 we obtain

characteristic point 0

**Example** 

displacement for thin shells)

 

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

For the analytic continuation of meromorphic solutions to the region and acceleration of the convergence we use 2-D PAs. To do this, the resulting series are reconstructed in rational

> 1 1 2 2 0 0 0 0

 

*m n m n*

*j k j k*

*<sup>i</sup> u P P*

<sup>00</sup>

*QP Q*

. *j j k k ijk ijk*

(which usually corresponds to the point of maximum transverse

.

0

0 *<sup>j</sup> <sup>k</sup>*

*ijk ijk i*

Since the proposed method is modified, all functions belonging to the boundary value problems can be expanded in powers of the independent variables and parameters. For

> 3 1 4 2 0 0 0 0

*m n m n*

*j k j k <sup>i</sup> u PP QP* 

can get the estimation of critical point localization by solving equations of the form

,

 

> 4 2 0 0 min : 0 *<sup>j</sup> <sup>k</sup> ijk <sup>i</sup> m n*

*j k P QP*

*m n*

*j k*

In practice this equation is transformed to its counterpart simpler form with respect to its

*ijk*

Let us consider the computational aspects of the proposed approach. We consider three types of PAs with respect to the independent variable, on the specified parameters, and 2-D. A typical behavior of the approximations for the BVP is governed by the following problem

.

*Q P* 

Rational functions have singular points which are determined by equating of the denominator to zero. According to the theory of bifurcation of solutions of ordinary differential equations, at these points either bifurcations or limit states are achieved. So we

, 1, *j j k k*

$$z = \sum\_{i=0}^{\infty} z\_i \varepsilon\_1^i. \tag{14}$$

This way of introducing the parameter leads to a system of successive approximations of Adomian decomposition method (ADM, see [5,6]). Substituting power series (4) into Eq. (3) and splitting it with respect to the powers of 1 , yields

$$\begin{aligned} \varepsilon\_1^0: \quad \dot{z}\_0 &= \frac{1}{\varepsilon}, \quad z\_0(0) = 0 \quad \Rightarrow \quad z\_0 = \frac{x}{\varepsilon}; \\\\ \varepsilon\_1^1: \quad \dot{z}\_1 &= -\frac{z\_0}{\varepsilon} = -\frac{x}{\varepsilon^2}, \quad z\_1(0) = 0 \quad \Rightarrow \quad z\_1 = -\frac{x^2}{2\varepsilon^2} = (-1)^1 \frac{1}{2!} \left(\frac{x}{\varepsilon}\right)^2; \\\\ \varepsilon\_1^2: \quad \dot{z}\_2 &= -\frac{z\_1}{\varepsilon} = -\frac{x^2}{\varepsilon^3}, \quad z\_2(0) = 0 \quad \Rightarrow \quad z\_2 = \frac{x^3}{6\varepsilon^3} = (-1)^2 \frac{1}{3!} \left(\frac{x}{\varepsilon}\right)^3; \\\\ \varepsilon\_1^n: \quad \dot{z}\_n &= -\frac{z\_{n-1}}{\varepsilon} = (-1)^n \frac{1}{(n-1)!} \left(\frac{x}{\varepsilon}\right)^n \left(\frac{1}{\varepsilon}\right), \quad z\_n(0) = 0 \quad \Rightarrow \quad z\_n = (-1)^n \frac{1}{n!} \left(\frac{x}{\varepsilon}\right)^{n+1}; \\\\ z &= \frac{x}{\varepsilon} - \frac{x^2}{2\varepsilon^2} \varepsilon\_1 + \frac{x^3}{6\varepsilon^3} \varepsilon\_1^2 + ... + (-1)^n \frac{1}{n!} \left(\frac{x}{\varepsilon}\right)^{n+1} \varepsilon\_1^n + .... \end{aligned}$$

For 1 1 one gets power series (2), which corresponds to the results reported in [5]. To accelerate the convergence we use 1-D and 2-D PAs. One can use 1-D PAs [1/1] ( <sup>1</sup> ( ) *z* , when *x const* 0 and ( ) , *<sup>x</sup> <sup>z</sup>* when 1 *const* 0 ) or 2-D PAs [(1,1)/(1,1)] ( <sup>1</sup> ( ,) *<sup>x</sup> z* ). Using PAs one obtains (for 1 )

$$z^{(c\_1)} = \frac{\chi}{\varepsilon} \left( 1 - \frac{3\chi}{6\varepsilon + 2\chi} \right) \tag{15}$$

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

 

 

1 ' '

2

 

*x*

2 2

(0) 0 (1 ) 3(1 ) (1 ) ; 4! 2 <sup>2</sup>

*z z x*

(1 ) (1 ) ; 3!

*z xx*

We obtain the HAM approximation in the following form

 

2 ' '

3 ' '

 

3

*x*

3

4 4

1 one obtains

Let us introduce parameter 1

14 3 3

*z x*

For 1 

1.

results:

11 0 0 1 1

 

: (1 ) (1 ) (1 )( (1 )), (0) 0 <sup>2</sup>

: 1 (1 ) 1, (0) 0 *z z z z zx* ;

12 1 1 <sup>2</sup> <sup>2</sup> : (1 ) (1 ), (0) 0 (1 ) ; 2!

3 2 4 ' ' 2 2 2

 

*x x*

1 1 <sup>1</sup> (1 ) (1 ) (1 ) ... . 2! 3!

 

1 1 (1 (1 ) (1 ) (1 ) ...) (1 ) 3(1 ) ... 2! <sup>2</sup>

Eq. (18) coincides with Eq. (12) after expanding the coefficients of Eq. (12) in the vicinity of

For the obtained approximations we use PAs as described above and get the following

1 1 ( ) ( ,) <sup>2</sup> , <sup>2</sup> *<sup>x</sup> x*

> 

( ) 2 2 . 2 2 *<sup>x</sup> x*

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

 

*z z*

*z*

in such a way that

*z* 

*z xx*

: (1 ) (1 ) (1 ) (1 ) 2(1 ) (1 ) , 6 2

2 3

2 2 23

   

 

(19)

 

2 3 2 2

*x*

2

*x*

*x x x*

*zz zx z z x*

 

13 2 2 3

43 2

 

*xx x*

2 3

*x x*

 

4

 

*x*

3

*x*

1 1 (1 ) ... ... . 3! 2 4!

 

*zz z x x z*

*zz z x x x*

Stability Calculation and Experimental Justification 11

2

(18)

*x*

$$z^{(\mathbf{x})} = z^{(\mathbf{c}\_1, \mathbf{x})} = \frac{2\mathbf{x}}{2\mathbf{c} + \mathbf{x}} \,. \tag{16}$$

R.h.s. of Eq. (15) contains singularity at point 0 in contrast to Eqs. (16). Thus, Eqs. (16) give the approximation with uniform accuracy when *x* grows.

Let us rewrite Eq. (11) in the following form

$$z' = \varepsilon\_1 (1 - z + (1 - \varepsilon)z'). \tag{17}$$

After substitution of the power series (14) in Eq. (17) one obtains a successive approximation of the homotopy analysis method (HAM) in the so called homotopy perturbation form (HPM, see [7])

$$z\_1^0: \quad z\_0^\cdot = 0, \quad z\_0(0) = 0 \quad \Longrightarrow \quad z\_0 = 0;$$

Applications of 2D Padé Approximants in Nonlinear Shell Theory: Stability Calculation and Experimental Justification 11

$$z\_1^1: \quad z\_1^\cdot = 1 - z\_0 + (1 - \varepsilon)z\_0^\cdot = 1, \quad z\_1(0) = 0 \quad \Longrightarrow \quad z\_1 = \infty$$

$$z\_1^2: \quad \dot{z\_2} = -z\_1 + (1 - \varepsilon)\dot{z\_1} = -\infty + (1 - \varepsilon), \quad z\_2(0) = 0 \quad \Rightarrow \quad z\_2 = -\frac{\chi^2}{2!} + (1 - \varepsilon)\chi;$$

$$\begin{aligned} x\_1^3: \quad \dot{z}\_3 &= -z\_2 + (1 - \varepsilon)z\_2^\cdot = \frac{\chi^2}{2} - (1 - \varepsilon)\chi + (1 - \varepsilon)(-\chi + (1 - \varepsilon)), \quad z\_3(0) = 0 \quad \Rightarrow\\ \implies \quad z\_3 &= \frac{\chi^3}{3!} - (1 - \varepsilon)\chi^2 + (1 - \varepsilon)^2\chi \end{aligned}$$

$$\begin{aligned} z\_4^4: \quad \dot{z\_4} &= -z\_3 + (1 - \varepsilon)\dot{z\_3} = -\frac{\chi^3}{6} + (1 - \varepsilon)\chi^2 - (1 - \varepsilon)^2\chi + (1 - \varepsilon)\left(\frac{\chi^2}{2} - 2(1 - \varepsilon)\chi + (1 - \varepsilon)^2\right) \\ z\_4(0) &= 0 \quad \Rightarrow \quad z\_4 = -\frac{\chi^4}{4!} + (1 - \varepsilon)\frac{\chi^3}{2} - 3(1 - \varepsilon)^2\frac{\chi^2}{2} + (1 - \varepsilon)^3\chi \end{aligned}$$

We obtain the HAM approximation in the following form

$$\pi z = \pi \varepsilon\_1 + \left( -\frac{\varkappa^2}{2!} + (1 - \varepsilon)\varkappa \right) \varepsilon\_1^2 + \left( \frac{\varkappa^3}{3!} - (1 - \varepsilon)\varkappa^2 + (1 - \varepsilon)^2 \varkappa \right) \varepsilon\_1^3 + \dots \dots$$

For 1 1 one obtains

10 Nonlinearity, Bifurcation and Chaos – Theory and Applications

and splitting it with respect to the powers of 1

'1

*x const* 0 and ( ) , *<sup>x</sup> <sup>z</sup>* when 1

1 ) *z*

R.h.s. of Eq. (15) contains singularity at point

Let us rewrite Eq. (11) in the following form

1

For 1 

obtains (for

(HPM, see [7])

0 '

*z zz*

*z zz*

*z*

*z z*

give the approximation with uniform accuracy when *x* grows.

0 '

This way of introducing the parameter leads to a system of successive approximations of Adomian decomposition method (ADM, see [5,6]). Substituting power series (4) into Eq. (3)

> , yields

10 0 0 <sup>1</sup> : , (0) 0 ;

<sup>1</sup> : , (0) 0 ( 1) ; <sup>2</sup> 2! *z x x x*

<sup>1</sup> : , (0) 0 ( 1) ; <sup>6</sup> 3! *z x xx*

1 1 <sup>1</sup> : ( 1) , (0) 0 ( 1) ; ( 1)! !

1 2 3 2

*<sup>n</sup> xx x n n <sup>x</sup>*

2 3 1 1 1 <sup>1</sup> ... ( 1) ... . 2 6 !

1 one gets power series (2), which corresponds to the results reported in [5]. To

*const* 0 ) or 2-D PAs [(1,1)/(1,1)] ( <sup>1</sup> ( ,) *<sup>x</sup> z*

6 2

2

<sup>1</sup> *z zz* ' (1 (1 ) ').

After substitution of the power series (14) in Eq. (17) one obtains a successive approximation of the homotopy analysis method (HAM) in the so called homotopy perturbation form

10 0 0

: 0, (0) 0 0; *zz z*

 

  *x*

*x*

*z x x*

*zz z*

1 ' 0 1 1 1 2 2 1 1

2 ' 1 2 1 2 3 3 2 2

*n n n n n n n*

*z z z*

 

> 

accelerate the convergence we use 1-D and 2-D PAs. One can use 1-D PAs [1/1] ( <sup>1</sup> ( )

<sup>1</sup> ( ) <sup>3</sup> <sup>1</sup>

<sup>1</sup> ( ) ( ,) 2

*<sup>x</sup> <sup>x</sup> x*

*x x*

 

 *x*

 

 

*n n*

3 2 3

*n n*

*n*

 

2 2

 

 

, (15)

. (16)

0 in contrast to Eqs. (16). Thus, Eqs. (16)

(17)

1

*z* , 

). Using PAs one

when

$$\begin{aligned} z &= \left(1 + (1 - \varepsilon) + (1 - \varepsilon)^2 + (1 - \varepsilon)^3 + \dots \right) \mathbf{x} + \left(-\frac{1}{2!} + (1 - \varepsilon) - 3(1 - \varepsilon)^2 \frac{1}{2} + \dots \right) \mathbf{x}^2 + \\ &+ \left(\frac{1}{3!} + (1 - \varepsilon)\frac{1}{2} + \dots \right) \mathbf{x}^3 - \frac{\mathbf{x}^4}{4!} + \dots \end{aligned} \tag{18}$$

Eq. (18) coincides with Eq. (12) after expanding the coefficients of Eq. (12) in the vicinity of 1.

For the obtained approximations we use PAs as described above and get the following results:

$$
\boldsymbol{z}^{\{\mathcal{E}\_1\}} = \boldsymbol{z}^{\{\mathcal{E}\_1, \boldsymbol{x}\}} = \frac{2\boldsymbol{x}}{2\boldsymbol{\varepsilon} + \boldsymbol{x}}',
$$

$$
\boldsymbol{z}^{\{\boldsymbol{x}\}} = \frac{2\left(2 - \boldsymbol{\varepsilon}\right)^2 \boldsymbol{x}}{2\left(2 - \boldsymbol{\varepsilon}\right) + \boldsymbol{x}}.
$$

Let us introduce parameter 1 in such a way that

$$z^\prime = \varepsilon\_1 \frac{1-z}{z}.\tag{19}$$

After substitution of the power series (14) in Eq. (19), one obtains a new system of successive approximations:

Applications of 2D Padé Approximants in Nonlinear Shell Theory:

whose coefficients are given depending on the variable for

**Figure 2.** The exact solution (solid line) of Eq. (20) for

<sup>2</sup> ' , 1 1, 0 1, 0. *zz x z x*

The graphs show that the solution is well described by the HAM approximation and MHAM-Padé «in average», and badly – in the boundary layer. The ADM approximation and MADM-Padé, on the contrary, is in good agreement with the behavior of solution in the vicinity of zero and in the bad one – on the stationary part. At the same time, 1-D and 2-D PAs, based on approximations of the MMPC, well describe the solution in the whole

The proposed MMPC method has been applied to calculate the deformation and stability of a long flexible elastic circular cylindrical shell of radius *R* with half the central angle 0

the case of cylindrical bending under uniform external pressure with a simple support of the longitudinal edges. The corresponding system of resolving equations in the normal form is given in [8]. Dependences of "dimensionless intensity of pressure *P* – deflection / *w R* " for

Fig. 3 shows the graphs of approximations for strongly non-linear BVP of the form

**5. Calculation of nonlinear deformation and stability of shells** 

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

in Fig. 2.

ADM, 2 – <sup>1</sup> *z*

HAM).

interval.

Stability Calculation and Experimental Justification 13

= 0.2 and approximate solutions (1 – three terms

in

(21)

= 0.2. Its graphs are presented

$$z\_1^0 \colon \quad z\_0^\cdot = 0 \,, \quad z\_0(0) = 0 \quad \Longrightarrow \quad z\_0 = 0 \colon$$

$$z\_1^1: \quad z\_1^\cdot = \frac{1 - z\_0}{\varepsilon} = \frac{1}{\varepsilon}, \quad z\_1(0) = 0 \quad \Rightarrow \quad z\_1 = \frac{\chi}{\varepsilon} = (-1)^2 \frac{1}{1!} \left(\frac{\chi}{\varepsilon}\right)^1;$$

$$z\_1^2: \quad \stackrel{\cdot}{z\_2} = -\frac{z\_1}{\varepsilon} = -\frac{\chi}{\varepsilon^2}, \quad z\_2(0) = 0 \quad \Rightarrow \quad z\_2 = -\frac{\chi^2}{2\varepsilon^2} = (-1)^3 \frac{1}{2!} \left(\frac{\chi}{\varepsilon}\right)^2;$$

$$z\_1^n \colon \quad \stackrel{\cdot}{z\_n} = -\frac{z\_{n-1}}{\varepsilon} = (-1)^{n+1} \frac{1}{(n-1)!} \left(\frac{\text{x}}{\varepsilon}\right)^{n-1} \left(\frac{1}{\varepsilon}\right), \quad z\_2(0) = 0 \quad \Rightarrow \quad z\_n = (-1)^{n+1} \frac{1}{n!} \left(\frac{\text{x}}{\varepsilon}\right)^n;$$

$$z = \frac{\chi}{\varepsilon} \varepsilon\_1 - \frac{1}{2!} \left(\frac{\chi}{\varepsilon}\right)^2 \varepsilon\_1^2 + \frac{1}{3!} \left(\frac{\chi}{\varepsilon}\right)^3 \varepsilon\_1^3 + \dots$$

For 1 1 one obtains PAs as follows:

$$z^{(\varepsilon\_1)} = z^{(\varepsilon)} = z^{(\varepsilon\_1, x)} = \frac{2x}{2\varepsilon + x} \dots$$

The coincidence of these approximations demonstrates their proximity to a unique function representing the exact solution in the fractional-rational form.

The ADM approximation describes well the exact solution only for a distance which is comparable with the value of natural small parameter . Despite the fact that the error of solutions of HAM is substantially less than the ADM, HAM does not accurately reflect the nature of solutions, namely the phenomenon of boundary layer in the vicinity of zero. At the same time, PAs for the ADM approximations for independent variable and PAs for the MMPC (1-D and 2-D) give satisfactory qualitative and quantitative results.

Similar results are provided by the analysis of approximations of BVP of the following problem

$$\begin{aligned} \varepsilon z' + \varkappa z &= \mathbf{x}, \\ \varepsilon (0) = 2, \quad 0 < \varepsilon << 1, \quad \varkappa \ge 0, \end{aligned} \tag{20}$$

whose coefficients are given depending on the variable for = 0.2. Its graphs are presented in Fig. 2.

12 Nonlinearity, Bifurcation and Chaos – Theory and Applications

1 one obtains PAs as follows:

0 '

approximations:

For 1 

problem

After substitution of the power series (14) in Eq. (19), one obtains a new system of successive

10 0 0

1 ' 0 2

2 ' 1 3 1 2 2 2 2 2

1 ' 1 1 1

*xx x*

*n n n n n n*

*z z z*

*z zz*

MMPC (1-D and 2-D) give satisfactory qualitative and quantitative results.

1 1 1 1

*z zz*

 

*z zz*

1 2

*z*

representing the exact solution in the fractional-rational form.

comparable with the value of natural small parameter

: 0, (0) 0 0; *zz z*

<sup>1</sup> 1 1 : , (0) 0 ( 1) ; 1! *z x x*

<sup>1</sup> : , (0) 0 ( 1) ; <sup>2</sup> 2! *z x x x*

1 1 <sup>1</sup> : ( 1) , (0) 0 ( 1) ; ( 1)! !

 

11 1 1 1 ... . 2! 3!

1 1 ( ) ( ) ( ,) 2

The coincidence of these approximations demonstrates their proximity to a unique function

The ADM approximation describes well the exact solution only for a distance which is

solutions of HAM is substantially less than the ADM, HAM does not accurately reflect the nature of solutions, namely the phenomenon of boundary layer in the vicinity of zero. At the same time, PAs for the ADM approximations for independent variable and PAs for the

Similar results are provided by the analysis of approximations of BVP of the following

' , (0) 2, 0 1, 0, *z xz x z x* 

 

 

*<sup>x</sup> <sup>x</sup> x*

.

*z x x*

2 3 2 3

> 

2

(20)

*n n*

 

*x*

 

 

*n n*

1

 

. Despite the fact that the error of

 

2 2

**Figure 2.** The exact solution (solid line) of Eq. (20) for = 0.2 and approximate solutions (1 – three terms ADM, 2 – <sup>1</sup> *z* for ADM, 3 – three terms HAM, 4 – *x z* for HAM, 5 – 2-D PAs for MMPC, ADM and HAM).

Fig. 3 shows the graphs of approximations for strongly non-linear BVP of the form

$$\begin{aligned} \varepsilon z' &= z^2 + \chi, \\ z(1) &= 1, \quad 0 < \varepsilon << 1, \quad \quad \chi \ge 0. \end{aligned} \tag{21}$$

The graphs show that the solution is well described by the HAM approximation and MHAM-Padé «in average», and badly – in the boundary layer. The ADM approximation and MADM-Padé, on the contrary, is in good agreement with the behavior of solution in the vicinity of zero and in the bad one – on the stationary part. At the same time, 1-D and 2-D PAs, based on approximations of the MMPC, well describe the solution in the whole interval.
