3. Dynamic Cosserat plate theory

In this section we review our stress, couple stress and kinematic assumptions of the Cosserat plate [7]. We consider the thin plate P, where h is the thickness of the plate and x<sup>3</sup> ¼ 0 represents its middle plane. The sets T and B are the top and bottom surfaces contained in the planes x<sup>3</sup> ¼ h=2, x<sup>3</sup> ¼ �h=2 respectively and the curve Γ is the boundary of the middle plane of the plate.

The set of points <sup>P</sup> <sup>¼</sup> <sup>Γ</sup> � � <sup>h</sup> 2 ; h 2 � � � � ∪ T ∪ B forms the entire surface of the plate and <sup>Γ</sup><sup>u</sup> � � <sup>h</sup> <sup>2</sup> , <sup>h</sup> 2 � � is the lateral part of the boundary where displacements and microrotations are prescribed. The notation Γ<sup>σ</sup> ¼ Γ\Γ<sup>u</sup> of the remainder we use to describe the lateral part of the boundary edge <sup>Γ</sup><sup>σ</sup> � � <sup>h</sup> <sup>2</sup> , <sup>h</sup> 2 � � where stress and couple stress are prescribed. We also use notation P<sup>0</sup> for the middle plane internal domain of the plate.

In our case we consider the vertical load and pure twisting momentum boundary conditions at the top and bottom of the plate, which can be written in the form:

$$
\sigma\_{33}(\mathbf{x}\_1, \mathbf{x}\_2, h/2, t) = \sigma^t(\mathbf{x}\_1, \mathbf{x}\_2, t), \\
\sigma\_{33}(\mathbf{x}\_1, \mathbf{x}\_2, -h/2, t) = \sigma^b(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{30}
$$

$$
\sigma\_{3\emptyset}(\varkappa\_1, \varkappa\_2, \pm h/2, t) = 0,\tag{31}
$$

$$
\mu\_{33}(\mathbf{x}\_1, \mathbf{x}\_2, h/2, t) = \mu^t(\mathbf{x}\_1, \mathbf{x}\_2, t), \\
\mu\_{33}(\mathbf{x}\_1, \mathbf{x}\_2, -h/2, t) = \mu^b(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{32}
$$

$$
\mu\_{3\emptyset}(\mathbf{x}\_1, \mathbf{x}\_2, \pm h/2, \mathbf{t}) = \mathbf{0},\tag{33}
$$

where ð Þ x1; x<sup>2</sup> ∈ P0:

We will also consider the rotatory inertia J in the form

$$\mathbf{J} = \begin{pmatrix} J\_{11} & J\_{12} & \mathbf{0} \\ J\_{12} & J\_{22} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & J\_{33} \end{pmatrix}.$$

Let A denote the set of all admissible states that satisfy the Cosserat plate straindisplacement relation Eq. (5) and let Θ be a functional on A defined by

$$\Theta(s,\eta) = U\_K^S + T\_C^S - \int\_{\Gamma\_0} \left( \mathcal{S} \cdot \mathcal{E} + \mathbf{P} \cdot \frac{\partial \mathcal{U}}{\partial t} - \hat{P} \cdot \mathcal{W} + \nu \mathfrak{Q}\_3^0 \right) da + \int\_{\Gamma\_s} \mathcal{S}\_o \cdot (\mathcal{U} - \mathcal{U}\_o) ds + \int\_{\Gamma\_s} \mathcal{S}\_n \cdot \mathcal{U} ds,\tag{34}$$

Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

for every <sup>s</sup> <sup>¼</sup> ½ � <sup>U</sup>; <sup>E</sup>; <sup>S</sup> <sup>∈</sup> <sup>A</sup>: Here <sup>P</sup>^ <sup>¼</sup> <sup>p</sup>^1; <sup>p</sup>^<sup>2</sup> � � and <sup>W</sup> <sup>¼</sup> <sup>W</sup>;<sup>W</sup> <sup>∗</sup> ð Þ, <sup>p</sup>^<sup>1</sup> <sup>¼</sup> <sup>η</sup><sup>p</sup> and <sup>p</sup>^<sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>3</sup> ð Þ 1 � η p

Here the plate stress and kinetic energy density by the formulas

$$U\_K^S = \int\_{P\_0} \Phi(\mathcal{S}) da , \, T\_K^S = \int\_{P\_0} \Upsilon\_C \left(\frac{\partial \mathcal{U}}{\partial t}\right) da \tag{35}$$

where P<sup>0</sup> is the internal domain of the middle plane of the plate.

<sup>Φ</sup>ð Þ¼� <sup>S</sup> <sup>3</sup>λð Þ <sup>M</sup>αα <sup>M</sup>ββ � � h3 μð Þ 3λ þ 2μ þ <sup>3</sup>ð Þ <sup>α</sup> <sup>þ</sup> <sup>μ</sup> <sup>M</sup><sup>2</sup> αβ 2h<sup>3</sup> αμ þ 3ð Þ α þ μ 160h<sup>3</sup> αμ <sup>8</sup>Q^ <sup>α</sup>Q^ <sup>α</sup> <sup>þ</sup> <sup>15</sup>QαQ^ <sup>α</sup> <sup>þ</sup> <sup>20</sup>Q^ <sup>α</sup><sup>Q</sup> <sup>∗</sup> <sup>α</sup> <sup>þ</sup> <sup>8</sup><sup>Q</sup> <sup>∗</sup> <sup>α</sup> Q <sup>∗</sup> α h i þ <sup>3</sup>ð Þ <sup>α</sup> � <sup>μ</sup> <sup>M</sup><sup>2</sup> αβ 2h<sup>3</sup> αμ <sup>þ</sup> <sup>α</sup> � <sup>μ</sup> 280h<sup>3</sup> αμ <sup>21</sup>Q<sup>α</sup> <sup>5</sup>Q^ <sup>α</sup> <sup>þ</sup> <sup>4</sup><sup>Q</sup> <sup>∗</sup> α h i � � � <sup>γ</sup> � <sup>ε</sup> 160hγε 24R<sup>2</sup> αα <sup>þ</sup> <sup>45</sup>R<sup>∗</sup> αα <sup>þ</sup> <sup>60</sup>RαβR<sup>∗</sup> αβ <sup>þ</sup> <sup>48</sup>R12R<sup>21</sup> h i þ <sup>3</sup>ð Þ <sup>γ</sup> <sup>þ</sup> <sup>ε</sup> <sup>S</sup> <sup>∗</sup> <sup>α</sup> S <sup>∗</sup> α 2h<sup>3</sup> γε <sup>þ</sup> <sup>γ</sup> <sup>þ</sup> <sup>ε</sup> 160h<sup>3</sup> γε 8R<sup>2</sup> αβ <sup>þ</sup> <sup>15</sup>R<sup>∗</sup> αβR<sup>∗</sup> αβ <sup>þ</sup> <sup>20</sup>RαβR<sup>∗</sup> αβ h i � <sup>3</sup><sup>β</sup> <sup>80</sup>hγð Þ <sup>3</sup><sup>β</sup> <sup>þ</sup> <sup>2</sup><sup>γ</sup> �8ð Þ <sup>R</sup>αα <sup>R</sup>ββ � � � <sup>15</sup> <sup>R</sup><sup>∗</sup> αα � � <sup>R</sup><sup>∗</sup> ββ � � � <sup>20</sup>ð Þ <sup>R</sup>αα <sup>R</sup><sup>∗</sup> αα � � h i � <sup>β</sup> 4γð Þ 3β þ 2γ <sup>2</sup>Rαα <sup>þ</sup> <sup>3</sup>R<sup>∗</sup> αα � �<sup>t</sup> � h V<sup>2</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> � � � � þ λ 560hμð Þ 3λ þ 2μ 5 þ 3η ð Þ <sup>1</sup> <sup>þ</sup> <sup>η</sup> pMαα � � <sup>þ</sup> ð Þ <sup>λ</sup> <sup>þ</sup> <sup>μ</sup> <sup>h</sup> 840μð Þ 3λ þ 2μ <sup>140</sup> <sup>þ</sup> <sup>168</sup><sup>η</sup> <sup>þ</sup> <sup>51</sup>η<sup>2</sup> 4 1ð Þ þ η 2 !<sup>p</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>λ</sup> <sup>þ</sup> <sup>μ</sup> <sup>h</sup> <sup>2</sup>μð Þ <sup>3</sup><sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>σ</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>ε</sup><sup>h</sup> 12hγε <sup>3</sup>T<sup>2</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup> � � � � (36)

and

$$\begin{split} \Upsilon\_{\mathbb{C}} \left( \frac{\partial \mathcal{U}}{\partial t} \right) &= \frac{h\rho}{2} \left( \frac{\partial \mathcal{W}}{\partial t} \right)^{2} + \frac{4h\rho}{15} \left( \frac{\partial \mathcal{W}^{\*}}{\partial t} \right)^{2} + \frac{2h\rho}{3} \left( \frac{\partial \mathcal{W}}{\partial t} \frac{\partial \mathcal{W}^{\*}}{\partial t} \right) + \frac{h^{3}\rho}{24} \left( \frac{\partial \Psi\_{a}}{\partial t} \right)^{2} \\ &+ \frac{4hI\_{a\beta}}{15} \left( \frac{\partial \mathcal{Q}^{0}\_{a}}{\partial t} \frac{\partial \mathcal{Q}^{0}\_{\beta}}{\partial t} \right) + \frac{hI\_{a\beta}}{2} \left( \frac{\partial \hat{\Omega}^{0}\_{a}}{\partial t} \frac{\partial \hat{\Omega}^{0}\_{\beta}}{\partial t} \right) + \frac{2hI\_{a\beta}}{3} \left( \frac{\partial \mathcal{Q}^{0}\_{a}}{\partial t} \frac{\partial \hat{\Omega}^{0}\_{\beta}}{\partial t} \right) + \frac{hI\_{33}}{6} \left( \frac{\partial \Omega\_{3}}{\partial t} \right)^{2} . \end{split}$$

S, U and E are the Cosserat plate stress, displacement and strain sets

$$\mathcal{S} = \left[ \mathbf{M}\_{a\beta}, \mathbf{Q}\_a, \mathbf{Q}\_a^\*, \hat{\mathbf{Q}}\_a, \mathbf{R}\_{a\beta}, \mathbf{R}\_{a\beta}^\*, \mathbf{S}\_{\beta}^\* \right], \tag{37}$$

$$\mathcal{S}\_n = \left[ \check{M}\_a, \check{\mathbf{Q}}^\*, \check{\mathbf{Q}}\_a, \check{\mathbf{R}}\_a, \check{\mathbf{R}}\_a^\*, \check{\mathbf{S}}^\* \right], \tag{38}$$

$$\mathcal{S}\_o = \left[ \Pi\_{oa}, \Pi\_{o3}, \Pi\_{o3}^\*, M\_{oa}, M\_{oa}^\*, M\_{o3}^\* \right], \tag{39}$$

$$\mathcal{U} = \left[ \Psi\_a, \mathcal{W}, \Omega\_3, \Omega\_a^0, \mathcal{W}^\*, \Omega\_a^0 \right], \tag{40}$$

$$\mathcal{E} = \left[ \mathfrak{e}\_{a\emptyset}, \mathfrak{o}\_{\beta}, \mathfrak{o}\_{a}^{\*}, \hat{\mathfrak{o}}\_{a}, \mathfrak{r}\_{\mathfrak{3}a}, \mathfrak{r}\_{a\beta}, \mathfrak{r}\_{a\beta}^{\*} \right],\tag{41}$$

where <sup>M</sup>αβn<sup>β</sup> <sup>¼</sup> <sup>Π</sup>o<sup>α</sup>, <sup>R</sup>αβn<sup>β</sup> <sup>¼</sup> Moα, <sup>Q</sup> <sup>∗</sup> <sup>α</sup> <sup>n</sup><sup>α</sup> <sup>¼</sup> <sup>Π</sup>o3, <sup>S</sup> <sup>∗</sup> <sup>α</sup> <sup>n</sup><sup>α</sup> <sup>¼</sup> <sup>M</sup><sup>∗</sup> <sup>o</sup>3, <sup>Q</sup>^ <sup>α</sup>n<sup>α</sup> <sup>¼</sup> <sup>Π</sup><sup>∗</sup> o3, R∗ αβn<sup>β</sup> <sup>¼</sup> <sup>M</sup><sup>∗</sup> <sup>o</sup><sup>α</sup>, <sup>M</sup>� <sup>α</sup> <sup>¼</sup> <sup>M</sup>αβnβ, <sup>Q</sup>� <sup>∗</sup> <sup>¼</sup> <sup>Q</sup> <sup>∗</sup> <sup>β</sup> <sup>n</sup>β, <sup>R</sup>�<sup>α</sup> <sup>¼</sup> <sup>R</sup>αβnβ, � <sup>S</sup> <sup>∗</sup> <sup>¼</sup> <sup>S</sup> <sup>∗</sup> <sup>β</sup> <sup>n</sup>β, <sup>Q</sup>�^ <sup>¼</sup> <sup>Q</sup>�^ <sup>β</sup>nβ, R� ∗ <sup>α</sup> <sup>¼</sup> <sup>R</sup>� <sup>∗</sup> αβnβ. (n<sup>β</sup> is the outward unit normal vector to Γu).

The plate characteristics provide the approximation of the components of the three-dimensional tensors σji and μji

$$
\sigma\_{a\beta} = \frac{\mathfrak{G}}{h^2} \zeta \mathcal{M}\_{a\beta}(\varkappa\_1, \varkappa\_2, t),
\tag{42}
$$

$$
\sigma\_{3\boldsymbol{\beta}} = \frac{3}{2h} \left( \mathbf{1} - \boldsymbol{\zeta}^2 \right) \mathbf{Q}\_{\boldsymbol{\beta}}(\boldsymbol{\omega}\_1, \boldsymbol{\omega}\_2, t), \tag{43}
$$

$$
\sigma\_{\beta 3} = \frac{3}{2h} \left( 1 - \zeta^2 \right) Q\_{\beta}^\*(\varkappa\_1, \varkappa\_2, t) + \frac{3}{2h} \hat{Q}\_{\beta}(\varkappa\_1, \varkappa\_2, t), \tag{44}
$$

$$
\sigma\_{33} = -\frac{3}{4} \left( \frac{1}{3} \zeta^3 - \zeta \right) p\_1(\varkappa\_1, \varkappa\_2, t) + \zeta p\_2(\varkappa\_1, \varkappa\_2, t) + \sigma\_0(\varkappa\_1, \varkappa\_2, t), \tag{45}
$$

$$
\mu\_{a\emptyset} = \frac{\mathfrak{Z}}{2\hbar} \left( \mathbf{1} - \zeta^2 \right) R\_{a\emptyset}(\mathbf{x}\_1, \mathbf{x}\_2, t) + \frac{\mathfrak{Z}}{2\hbar} R\_{a\emptyset}^\*(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{46}
$$

$$
\mu\_{\beta 3} = \frac{\mathsf{G}}{h^2} \zeta \mathsf{S}\_{\beta}^\*(\varkappa\_1, \varkappa\_2, t),
\tag{47}
$$

$$
\mu\_{3\beta} = \mathbf{0},
\tag{48}
$$

$$
\mu\_{33} = \zeta V(\varkappa\_1, \varkappa\_2, t) + T(\varkappa\_1, \varkappa\_2, t), \tag{49}
$$

where

$$p(\mathbf{x}\_1, \mathbf{x}\_2, t) = \sigma^t(\mathbf{x}\_1, \mathbf{x}\_2, t) - \sigma^b(\mathbf{x}\_1, \mathbf{x}\_2, t),\tag{50}$$

$$\sigma\_0(\mathbf{x}\_1, \mathbf{x}\_2, t) = \frac{1}{2} \left( \sigma^t(\mathbf{x}\_1, \mathbf{x}\_2, t) + \sigma^b(\mathbf{x}\_1, \mathbf{x}\_2, t) \right), \tag{51}$$

$$V(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) = \frac{1}{2} \left( \boldsymbol{\mu}^t(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) - \boldsymbol{\mu}^b(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) \right), \tag{52}$$

$$T(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) = \frac{1}{2} \left( \mu^t(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) + \mu^b(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t) \right). \tag{53}$$

The pressures p<sup>1</sup> and p<sup>2</sup> are chosen in the form

$$p\_1(\mathbf{x}\_1, \mathbf{x}\_2, t) = \eta p(\mathbf{x}\_1, \mathbf{x}\_2, t),\tag{54}$$

$$p\_2(\mathbf{x}\_1, \mathbf{x}\_2, t) = \frac{(\mathbf{1} - \eta)}{2} p(\mathbf{x}\_1, \mathbf{x}\_2, t). \tag{55}$$

and η∈ R is called splitting parameter.

The three-dimensional displacements ui and microrotations ϕ<sup>i</sup>

$$
\mu\_a = \frac{h}{2} \zeta \Psi\_a(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{56}
$$

$$
\mu\_3 = W(\varkappa\_1, \varkappa\_2, t) + \left(1 - \zeta^2\right) W^\*\left(\varkappa\_1, \varkappa\_2, t\right), \tag{57}
$$

$$
\phi\_a = \Omega\_a^0(\varkappa\_1, \varkappa\_2, t) \left(1 - \zeta^2\right) + \hat{\Omega}\_a(\varkappa\_1, \varkappa\_2, t), \tag{58}
$$

Distinctive Characteristics of Cosserat Plate Free Vibrations DOI: http://dx.doi.org/10.5772/intechopen.87044

$$
\phi\_3 = \zeta \Omega\_3(\varkappa\_1, \varkappa\_2, t),
\tag{59}
$$

and the three-dimensional strain and torsion tensors γji and χji

$$\gamma\_{a\beta} = \frac{6}{h^2} \zeta e\_{a\beta}(\varkappa\_1, \varkappa\_2, t), \tag{60}$$

$$\gamma\_{3\boldsymbol{\beta}} = \frac{3}{2h} \left( \mathbf{1} - \boldsymbol{\zeta}^2 \right) \alpha\_{\boldsymbol{\beta}}(\boldsymbol{\varkappa}\_1, \boldsymbol{\varkappa}\_2, t), \tag{61}$$

$$
\gamma\_{\beta 3} = \frac{3}{2h} \left( 1 - \zeta^2 \right) \alpha\_{\beta}^\*(\mathbf{x}\_1, \mathbf{x}\_2, t) + \frac{3}{2h} \hat{\alpha}\_{\beta}(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{62}
$$

$$\chi\_{a\boldsymbol{\beta}} = \frac{3}{2h} (\mathbf{1} - \boldsymbol{\zeta}^2) \tau\_{a\boldsymbol{\beta}}(\mathbf{x}\_1, \mathbf{x}\_2, t) + \frac{3}{2h} \tau\_{a\boldsymbol{\beta}}^\*(\mathbf{x}\_1, \mathbf{x}\_2, t), \tag{63}$$

$$
\chi\_{3\theta} = \frac{6}{h^2} \zeta \tau\_{\theta}^\*(\varkappa\_1, \varkappa\_2, t),
\tag{64}
$$

where <sup>ζ</sup> <sup>¼</sup> <sup>2</sup>x<sup>3</sup> h .

Then zero variation of the functional

$$
\delta\Theta(s,\eta) = 0
$$

is equivalent to the plate bending system of equations (A) and constitutive formulas (B) mixed problems.

A. The bending equilibrium system of equations:

$$\mathbf{M}\_{a\beta,a} - \mathbf{Q}\_{\beta} = I\_1 \frac{\partial^2 \Psi\_{\beta}}{\partial t^2},\tag{65}$$

$$Q\_{a,a}^{\*} + \hat{p}\_1 = I\_2 \frac{\partial^2 W^\*}{\partial t^2},\tag{66}$$

$$R\_{a\beta,a} + \varepsilon\_{3\beta\gamma} \left( Q\_{\gamma}^{\ast} - Q\_{\gamma} \right) = I\_{a\beta} \frac{\partial^2 \Omega\_a^0}{\partial t^2},\tag{67}$$

$$
\varepsilon\_{3\beta\gamma} \mathcal{M}\_{\beta\gamma} + \mathcal{S}\_{a,a}^\* = I\_3 \frac{\partial^2 \Omega\_3}{\partial t^2},\tag{68}
$$

$$
\hat{Q}\_{a,a} + \hat{p}\_2 = I\_2 \frac{\partial^2 W}{\partial t^2},
\tag{69}
$$

$$R^\*\_{a\beta,a} + \varepsilon\_{3\beta\gamma} \hat{\mathbf{Q}}\_{\gamma} = I^0\_{a\beta} \frac{\partial^2 \hat{\mathbf{Q}}^0\_a}{\partial t^2},\tag{70}$$

where <sup>I</sup><sup>1</sup> <sup>¼</sup> <sup>h</sup><sup>3</sup> <sup>12</sup> <sup>ρ</sup>, <sup>I</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup><sup>h</sup> <sup>3</sup> <sup>ρ</sup>, <sup>I</sup>αβ <sup>¼</sup> <sup>5</sup><sup>h</sup> <sup>6</sup> <sup>J</sup>αβ, <sup>I</sup><sup>3</sup> <sup>¼</sup> <sup>h</sup><sup>2</sup> <sup>6</sup> J33, I 0 αβ <sup>¼</sup> <sup>2</sup><sup>h</sup> <sup>3</sup> Jαβ, p^<sup>1</sup> ¼ ηoptp, and <sup>p</sup>^<sup>2</sup> <sup>¼</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup> � <sup>η</sup>opt <sup>p</sup>, with the resultant traction boundary conditions:

$$M\_{a\beta}n\_{\beta} = \Pi\_{\alpha\alpha} \quad R\_{a\beta}n\_{\beta} = M\_{\alpha\alpha} \tag{71}$$

$$\mathbf{Q}\_a^\* \mathfrak{n}\_a = \Pi\_{a\mathfrak{3}} \quad \mathbf{S}\_a^\* \mathfrak{n}\_a = \Upsilon\_{a\mathfrak{3}} \tag{72}$$

at the part Γ<sup>σ</sup> and the resultant displacement boundary conditions

$$
\Psi\_a = \Psi\_{oa}, \; W = W\_{o\nu} \Omega\_a^0 = \Omega\_{oa}^0, \; \Omega\_3 = \Omega\_{03}, \tag{73}
$$

at the part Γu:

B. Constitutive formulas in the reverse form:<sup>1</sup>

$$\mathcal{M}\_{\text{an}} = \frac{\mu(\lambda + \mu)h^3}{\Im(\lambda + 2\mu)} \Psi\_{a,a} + \frac{\lambda \mu h^3}{6(\lambda + 2\mu)} \Psi\_{\beta,\beta} + \frac{\left(3p\_1 + 5p\_2\right)\lambda h^2}{\Im 0(\lambda + 2\mu)},\tag{74}$$

$$M\_{\beta a} = \frac{(\mu - a)h^{\beta}}{12} \Psi\_{a,\beta} + \frac{(\mu + a)h^{\beta}}{12} \Psi\_{\beta,a} + \left(-1\right)^{a'} \frac{ah^{\beta}}{6} \Omega\_3,\tag{75}$$

$$R\_{\beta a} = \frac{\mathsf{S}(\mathsf{y} - \mathsf{e})h}{\mathsf{G}} \boldsymbol{\Omega}\_{\beta, a}^{0} + \frac{\mathsf{S}(\mathsf{y} + \mathsf{e})h}{\mathsf{G}} \boldsymbol{\Omega}\_{a, \beta^{\circ}}^{0} \tag{76}$$

$$R\_{aa} = \frac{10h\gamma(\beta+\gamma)}{\Re(\beta+2\gamma)}\Omega\_{a,a}^{0} + \frac{5h\beta\gamma}{\Im(\beta+2\gamma)}\Omega\_{\beta,\beta}^{0},\tag{77}$$

$$R^{\*}\_{\beta a} = \frac{\mathfrak{D}(\mathfrak{y} - \mathfrak{e})\hbar}{\mathfrak{B}} \hat{\mathfrak{Q}}\_{\beta, a} + \frac{\mathfrak{Z}(\mathfrak{y} + \mathfrak{e})\hbar}{\mathfrak{B}} \hat{\mathfrak{Q}}\_{a, \beta \mathfrak{e}} \tag{78}$$

$$R\_{aa}^{\*} = \frac{8\chi(\chi+\beta)h}{\Im(\beta+2\chi)}\hat{\Omega}\_{a,a} + \frac{4\chi\beta h}{\Im(\beta+2\chi)}\hat{\Omega}\_{\beta,\beta},\tag{79}$$

$$\mathcal{Q}\_a = \frac{\mathsf{5}(\mu + a)h}{6} \mathcal{W}\_a + \frac{\mathsf{5}(\mu - a)h}{6} \mathcal{W}\_{,a} + \frac{\mathsf{2}(\mu - a)h}{3} \mathcal{W}\_{,a}^\* + (-1)^\beta \frac{\mathsf{5}ha}{3} \mathcal{Q}\_\beta^0 + (-1)^\beta \frac{\mathsf{5}ha}{3} \hat{\Omega}\_{\beta \mu} \tag{80}$$

$$\mathcal{Q}\_a^\* = \frac{\mathsf{5}(\mu - a)\mathsf{h}}{\mathsf{6}} \Psi\_a + \frac{\mathsf{5}(\mu - a)^2 \mathsf{h}}{\mathsf{6}(\mu + a)} \mathcal{W}\_{,a} + \frac{2(\mu + a)\mathsf{h}}{\mathsf{3}} \mathcal{W}\_{,a}^\* + (-\mathsf{1})^a \frac{\mathsf{5}\mathsf{h} a}{\mathsf{3}} \left( \mathfrak{Q}\_{\not\equiv}^0 + \frac{(\mu - a)}{(\mu + a)} \hat{\Omega}\_{\not\equiv} \right), \tag{81}$$

$$
\hat{Q}\_a = \frac{8a\mu h}{3(\mu+a)} \mathcal{W}\_{,a} + (-1)^a \frac{8a\mu h}{3(\mu+a)} \hat{\Omega}\_\beta,\tag{82}
$$

$$\mathbf{S}\_a^\* = \frac{5\gamma e h^3}{\Im(\gamma + \varepsilon)} \boldsymbol{\Omega}\_{3,a} \tag{83}$$

and the optimal value ηopt of the splitting parameter is given as in [10]

$$\eta\_{\rm opt} = \frac{2\,\mathcal{W}^{(00)} - \mathcal{W}^{(10)} - \mathcal{W}^{(01)}}{2\left(\mathcal{W}^{(11)} + \mathcal{W}^{(00)} - \mathcal{W}^{(10)} - \mathcal{W}^{(01)}\right)}.\tag{84}$$

where

$$\mathcal{W}^{(\vec{\eta})} = \mathcal{S}|\_{\eta=\mathfrak{i}} \cdot \mathcal{E}|\_{\eta=\mathfrak{j}}.$$

We also assume that the initial condition can be presented in the form

$$\mathcal{U}(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{0}) = \mathcal{U}^0(\mathbf{x}\_1, \mathbf{x}\_2), \\ \frac{\partial \mathcal{U}}{\partial t}(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{0}) = \mathcal{V}^0(\mathbf{x}\_1, \mathbf{x}\_2).$$
