In the latter case, the problem defined on the cell (
2.1)–(
2.4) can be recast in the fast polar coordinates
\(r\,,\;\theta \), and it is governed by the following equations:
$$\begin{aligned}&\displaystyle \frac{\partial ^{2}u_{1}^{- \left( 01 \right) }}{\partial r^{2}}+\frac{1}{r}\cdot \frac{\partial u_{1}^{- \left( 01 \right) }}{\partial r}+\frac{1}{r^{2}}\cdot \frac{\partial ^{2}u_{1}^{- \left( 01 \right) }}{\partial \theta ^{2}}=0\quad \mathrm {in}\,\varOmega _{i}^{- \left( 0 \right) }; \end{aligned}$$
(3.1)
$$\begin{aligned}&\displaystyle \frac{\partial ^{2}u_{1}^{+ \left( 01 \right) }}{\partial r^{2}}+\frac{1}{r}\cdot \frac{\partial u_{1}^{+ \left( 01 \right) }}{\partial r}+\frac{1}{r^{2}}\cdot \frac{\partial ^{2}u_{1}^{+ \left( 01 \right) }}{\partial \theta ^{2}}=0\quad \mathrm {in}\,\varOmega _{i}^{+ \left( 0 \right) }; \end{aligned}$$
(3.2)
$$\begin{aligned}&\displaystyle u_{1}^{+ \left( 01 \right) }=u_{1}^{- \left( 01 \right) };\quad \frac{\partial u_{1}^{+ \left( 01 \right) }}{\partial r}-\lambda \frac{\partial u_{1}^{- \left( 01 \right) }}{\partial r}=\left( \lambda -1 \right) \left( \frac{\partial u_{0}}{\partial x}\cos \theta +\frac{\partial u_{0}}{\partial y}\sin \theta \right) \quad \mathrm {for}\,r=\tilde{a}; \end{aligned}$$
(3.3)
$$\begin{aligned}&\displaystyle u_{1}^{+ \left( 01 \right) }\rightarrow 0; \quad \frac{\partial u_{1}^{+ \left( 01 \right) }}{\partial r}\rightarrow 0\quad \mathrm {for}\,r\rightarrow \infty . \end{aligned}$$
(3.4)
A solution to the coupled problem (
3.1)–(
3.4) takes the form
$$\begin{aligned}&u_{1}^{-\;\left( {01} \right) } =A_{1}^{\left( {01} \right) } \,r\cos \,\theta +A_{2}^{\left( {01} \right) } \,r\sin \,\theta , \end{aligned}$$
(3.5)
$$\begin{aligned}&u_{1}^{+\;\left( {01} \right) } =\frac{B_{1}^{\left( {01} \right) } }{r}\,\cos \,\theta +\frac{B_{2}^{\left( {01} \right) } }{r}\,\sin \,\theta , \end{aligned}$$
(3.6)
where
\(A_{1}^{\left( {01} \right) } \,,\;A_{2}^{\left( {01} \right) } \,,\;B_{1}^{\left( {01} \right) } \,,\;B_{2}^{\left( {01} \right) } \) are arbitrary constants.
Observe that relations (
3.5), (
3.6) include four arbitrary constants, i.e. each two for them stand as the basic functions
\(\cos \,\theta \) and
\(\sin \,\theta \) (
\(A_{1}^{\left( {01} \right) } \,,\;B_{1}^{\left( {01} \right) } \) and
\(A_{2}^{\left( {01} \right) } \,,\;B_{2}^{\left( {01} \right) } \), respectively), which are defined through compatibility conditions (
3.3). Since the systems of equations which require definition of the integration constants
\(A_{1}^{\left( {01} \right) } \,,\;B_{1}^{\left( {01} \right) } \) and
\(A_{2}^{\left( {01} \right) } \,,\;B_{2}^{\left( {01} \right) } \) are the same to avoid repetitions, to avoid repetition, we present only one of them:
$$\begin{aligned} \left\{ {\begin{array}{l} A_{1}^{\left( {01} \right) } \,a=B_{1}^{\left( {01} \right) } \,a^{-\,1}, \\ -B_{1}^{\left( {01} \right) } \,a^{-\,2}-\lambda A_{1}^{\left( {01} \right) } =\frac{\partial u_{0} }{\partial x}\,\left( {\lambda -1} \right) . \\ \end{array}} \right. \end{aligned}$$
(3.7)
Solving equations (
3.7) allows to find the integration constants
$$\begin{aligned} \left\{ {\begin{array}{l} A_{1}^{\left( {01} \right) } =-\frac{\lambda -1}{\lambda +1}\,\frac{\partial u_{0} }{\partial x}=\frac{\partial u_{0} }{\partial x}\,A^{\left( {01} \right) \,^{*}}, \\ B_{1}^{\left( {01} \right) } =-\frac{\left( {\lambda -1} \right) \,a^{2}}{\lambda +1}\,\frac{\partial u_{0} }{\partial x}=\frac{\partial u_{0} }{\partial x}\,B^{\left( {01} \right) \,^{*}}, \\ \end{array}} \right. \end{aligned}$$
(3.8)
where
$$\begin{aligned} A^{\left( {01} \right) \,^{*}} =-\frac{\lambda -1}{\lambda +1}\,\quad \text{ and }\quad \;\;B^{\left( {01} \right) \,^{*}} =-\frac{\left( {\lambda -1} \right) \,a^{2}}{\lambda +1}. \end{aligned}$$
(3.9)
It is clear that for arbitrary constants
\(A_{2}^{\left( {01} \right) } \,\;\text{ and }\;\;B_{2}^{\left( {01} \right) } \) we have
$$\begin{aligned} A_{2}^{\left( {01} \right) } =A_{1}^{\left( {01} \right) } \,\quad \text{ and }\quad \;\;B_{2}^{\left( {01} \right) } =B_{1}^{\left( {01} \right) } \,\,\left( {\frac{\partial u_{0} }{\partial x}\rightarrow \frac{\partial u_{0} }{\partial y}} \right) . \end{aligned}$$
(3.10)
Therefore, the solution of the (01) approximation is as follows:
$$\begin{aligned} { u}_{{1}}^{{- }\left( {01} \right) }{=-}\frac{{\partial }{u}_{{0}}}{\partial {x}}\frac{{\lambda -1}}{{\lambda +1}}{r}\cos {\theta }{-}\frac{{\partial }{u}_{{0}}}{\partial {y}}\frac{{\lambda -1}}{{\lambda +1}}{r}\sin {\theta },\quad {u}_{{1}}^{{+ }\left( {01} \right) }{=-}\frac{{\partial }{u}_{{0}}}{\partial {x}}\frac{{\lambda -1}}{{\lambda +1}}{a}^{{2}}\frac{\cos {\theta }}{{r}}{-}\frac{{\partial }{u}_{{0}}}{\partial {y}}\frac{{\lambda -1}}{{\lambda +1}}{a}^{{2}}\frac{\sin {\theta }}{{r}},\nonumber \\ \end{aligned}$$
(3.11)
or equivalently
$$\begin{aligned} u_{1}^{- \left( 01 \right) }=-\frac{\lambda -1}{\lambda +1}\left( \frac{\partial u_{0}}{\partial x}\xi +\frac{\partial u_{0}}{\partial y}\eta \right) , \quad u_{1}^{+ \left( 01 \right) }=-\frac{\partial u_{0}}{\partial x}\frac{\lambda -1}{\lambda +1}a^{2}\left( \frac{\partial u_{0}}{\partial x}\frac{\xi }{\xi ^{2}+\eta ^{2}}+\frac{\partial u_{0}}{\partial y}\frac{\eta }{\xi ^{2}+\eta ^{2}} \right) . \end{aligned}$$
(3.11')
In what follows, we construct the (02) approximation of the SAM, which refers to the solution of the problem in the cell matrix
\(\Omega _{i}^{*} \).
Now, the periodicity conditions (
2.4) located on opposite sides of the cell are satisfied, and the compatibility conditions (
2.3) are ignored. Since the function
\(u_{1}^{\left( {02} \right) } \) should correct errors occurring in the solution
\(u_{1}^{+\;\left( {01} \right) } \) on the sizes of the cell, the following boundary value problem holds:
$$\begin{aligned}&\Delta u_{1}^{\left( {02} \right) } =0\quad \text{ in }\,\Omega _{i}^{*} ; \end{aligned}$$
(3.12)
$$\begin{aligned}&\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\xi =1} } \right. =\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\xi =\,-\,1} } \right. , \nonumber \\&{\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) }{\partial \xi }\,\,\left| {\,_{\xi =1} } \right. =\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) }{\partial \xi }\,\,\left| {\,_{\xi =\,-\,1} } \right. ,} \end{aligned}$$
(3.13)
$$\begin{aligned}&\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\eta =1} } \right. =\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\eta =\,-\,1} } \right. , \nonumber \\&\quad {\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) }{\partial \eta }\,\,\left| {\,_{\eta =1} } \right. =\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{1}^{\left( {02} \right) } } \right) }{\partial \eta }\,\,\left| {\,_{\eta =\,-\,1} } \right. }. \end{aligned}$$
(3.14)
We assume
$$\begin{aligned} u_{1}^{\left( {02} \right) } =u_{11}^{\left( {02} \right) } +u_{12}^{\left( {02} \right) } , \end{aligned}$$
(3.15)
where
\(u_{11}^{\left( {02} \right) } \) satisfies non-homogenous boundary conditions with regard to
\(\xi \) and homogenous boundary condition with regard to
\(\eta \). Therefore, the following equations should be satisfied:
$$\begin{aligned}&\Delta u_{11}^{\left( {02} \right) } =0\quad \text{ in }\, \Omega _{i}^{*} ; \end{aligned}$$
(3.16)
$$\begin{aligned}&\left( {u_{1}^{+\;\left( {01} \right) } +u_{11}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\xi =1} } \right. =\left( {u_{1}^{+\;\left( {01} \right) } +u_{11}^{\left( {02} \right) } } \right) \,\,\left| {\,_{\xi =\,-\,1} ,}\right. \nonumber \\&{\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{11}^{\left( {02} \right) } } \right) }{\partial \xi }\,\,\left| {\,_{\xi =1} } \right. =\frac{\partial \,\left( {u_{1}^{+\;\left( {01} \right) } +u_{11}^{\left( {02} \right) } } \right) }{\partial \xi }\,\,\left| {\,_{\xi =\,-\,1} } \right. } , \end{aligned}$$
(3.17)
$$\begin{aligned}&u_{11}^{\left( {02} \right) } \,\left| {\,_{\eta =1} } \right. =u_{11}^{\left( {02} \right) } \,\left| {\,_{\eta =\,-\,1} } \right. \,;\;\;\frac{\partial u_{11}^{\left( {02} \right) } }{\partial \eta }\,\left| {\,_{\eta =1} } \right. =\frac{\partial u_{11}^{\left( {02} \right) } }{\partial \eta }\,\left| {\,_{\eta =\,-\,1} } \right. . \end{aligned}$$
(3.18)
It is obvious that the function
\(u_{12}^{\left( {02} \right) } \) can be found in an analogous way simply by using the change:
\(\xi \,\,{\begin{array}{*{20}c} \rightarrow \\ \leftarrow \\ \end{array} }\,\,\eta \).
In order to satisfy the boundary conditions (
3.17), we recast them considering the (01) approximation of (
3.11) to the following form:
$$\begin{aligned} \begin{array}{l} \;u_{11}^{\left( {02} \right) } \,\left| {\,_{\xi =1} } \right. -u_{11}^{\left( {02} \right) } \,\left| {\,_{\xi =\,-\,1} } \right. =\frac{\partial u_{0} }{\partial x}\cdot \frac{2\,a^{2}}{1+\eta ^{2}}\,\frac{\lambda -1}{\lambda +1}, \\ \frac{\partial u_{11}^{\left( {02} \right) } }{\partial \xi }\,\left| {\,_{\xi =1} } \right. -\frac{\partial u_{11}^{\left( {02} \right) } }{\partial \xi }\,\left| {\,_{\xi =\,-\,1} } \right. =-\frac{\partial u_{0} }{\partial y}\cdot \frac{4\,a^{2}\eta }{\left( {1+\eta ^{2}} \right) ^{2}}\,\frac{\lambda -1}{\lambda +1}. \\ \end{array} \end{aligned}$$
(3.20)
The right-hand sides of equations (
3.20) are expand into the following Fourier series:
$$\begin{aligned} \begin{array}{l} \frac{1}{1+\eta ^{2}}=\frac{\pi }{4}+\sum \limits _{n=1}^\infty {\,\left[ {e^{-\,\pi n}\,\hbox {Im}\,E_{1} \left( {-\pi n+i\pi n} \right) {-e^{\pi n}\,\hbox {Im}\,E_{1} \left( {\pi n+i\pi n} \right) +\pi e^{-\,\pi n}}} \right] } \cos \,\pi n\eta , \\ \frac{\eta }{\left( {1+\eta ^{2}} \right) ^{2}}=\frac{\pi }{2}\sum \limits _{n=1}^\infty {\,n\,\left[ {e^{\pi n}\,\hbox {Im}\,E_{1} \left( {\pi n+i\pi n} \right) {-e^{-\,\pi n}\,\hbox {Im}\,E_{1} \left( {-\pi n+i\pi n} \right) -\pi e^{-\,\pi n}}} \right] } \,\sin \,\pi n\eta , \\ \end{array} \end{aligned}$$
(3.21)
where
\(i=\sqrt{-1} \) and
\(E_{1} \) stands for the exponential integral [
28].
Comparison of the corresponding coefficients in (
3.20), accounting for (
3.21) yields the (02) approximation coefficients:
$$\begin{aligned}&A_{0}^{\left( {02} \right) } =0\,;\;A_{n}^{\left( {02} \right) } =D_{n}^{\left( {02} \right) } =0\,,\;\;n=1\,,\;2,\ldots ; \nonumber \\&B_{0}^{\left( {02} \right) } =\frac{\partial u_{0} }{\partial x}\,B_{0}^{\left( {02} \right) \,^{*}} ;\;\;B_{n}^{\left( {02} \right) } =\frac{\partial u_{0} }{\partial x}\,B_{n}^{\left( {02} \right) \,^{*}} \,;\;\;C_{n}^{\left( {02} \right) } =\frac{\partial u_{0} }{\partial y}\,C_{n}^{\left( {02} \right) \,^{*}} \,,\;\;n=1\,,\;2\,,\;...\,; \end{aligned}$$
(3.22)
$$\begin{aligned}&B_{0}^{\left( {02} \right) \,^{*}} =\frac{\lambda -1}{\lambda +1}\cdot \frac{\pi a^{2}}{4}; \end{aligned}$$
(3.23)
$$\begin{aligned}&B_{n}^{\left( {02} \right) \,^{*}} =-C_{n}^{\left( {02} \right) \,^{*}} =\frac{\lambda -1}{\lambda +1}\,a^{2}\,S_{n} , \end{aligned}$$
(3.24)
where
$$\begin{aligned} {S}_{\mathrm{n}} =\frac{{e}^{-\,\uppi {n}}\,\hbox {Im}\,{E}_{1} \left( {-\uppi {n+i}\uppi {n}} \right) -{e}^{\uppi {n}}\,\hbox {Im}\,{E}_{1} \left( {\uppi {n}+\hbox {i}\uppi {n}} \right) +\uppi {e}^{-\,\uppi {n}}}{\sinh \,\uppi {n}}. \end{aligned}$$
(3.25)
Consequently, we get
$$\begin{aligned} {u}_{11}^{\left( {02} \right) } =\frac{\partial {u}_{0} }{\partial {x}}\,{B}_{0}^{\left( {02} \right) \,^{*}} \xi +\sum \limits _{{n}=1}^\infty {\,\left( {\frac{\partial {u}_{0} }{\partial {x}}\,{B}_{\mathrm{n}}^{\left( {02} \right) \,^{*}} \sinh \,\uppi {n}\xi \,\cos \,\uppi {n}\eta -\frac{\partial {u}_{0} }{\partial {y}}\,{C}_{\mathrm{n}}^{\left( {02} \right) \,^{*}} \cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } . \end{aligned}$$
(3.26)
Proceeding in an analogous way yields
$$\begin{aligned} u_{12}^{\left( {02} \right) } =u_{11}^{\left( {02} \right) } \;\;\left( {\frac{\partial u_{0} }{\partial x}\,\,{\begin{array}{*{20}c} \rightarrow \\ \leftarrow \\ \end{array} }\,\frac{\partial u_{0} }{\partial y}\,;\;\;\xi \,\,{\begin{array}{*{20}c} \rightarrow \\ \leftarrow \\ \end{array} }\,\,\eta } \right) . \end{aligned}$$
(3.27)
Finally, the (02) order approximation takes the following form:
$$\begin{aligned}&{u}_{1}^{\left( {02} \right) } =\frac{\lambda -1}{\lambda +1}\cdot \frac{\uppi {a}^{2}}{4}\left( {\frac{\partial {u}_{0} }{\partial {x}}\,\xi +\frac{\partial {u}_{0} }{\partial {y}}\,\eta } \right) +\frac{\lambda -1}{\lambda +1}\,{a}^{2}\sum \limits _{{n}=1}^\infty {\,{S}_{\mathrm{n}} \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\uppi {n}\xi \,\cos \,\uppi {n}\eta -} \right. } \right. } \nonumber \\&\quad -\left. {\left. {\cosh \,\uppi {n}\eta \,\sin \,\uppi {n}\xi } \right) +\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi {n}\eta \,\cos \,\uppi {n}\xi -\cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } \right] . \end{aligned}$$
(3.28)
In the (03) order approximation, we should remove the lack of compliance of the function
\(u_{1}^{\left( {02} \right) } \) governed by (
3.28), on the circular contour of an inclusion with radius
\(r=a\). For this purpose, we develop the function
\(u_{1}^{\left( {02} \right) } \) into a series regarding polar coordinates
\(r\,,\;\theta \); assuming a small radius
r of the inclusions, we obtain
$$\begin{aligned}&{B}_{0}^{\left( {02} \right) \,^{*}} \left( {\frac{\partial {u}_{0} }{\partial {x}}\,\xi +\frac{\partial {u}_{0} }{\partial {y}}\,\eta } \right) +\sum \limits _{{n}=1}^\infty {\,{B}_{\mathrm{n}}^{\left( {02} \right) \,^{*}} \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\uppi {n}\xi \,\cos \,\uppi {n}\eta -\cosh \,\uppi {n}\eta \,\sin \,\uppi {n}\xi } \right) } \right. } \\&\quad \left. {+\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi {n}\eta \,\cos \,\uppi {n}\xi -\cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } \right] = B_{0}^{\left( {02} \right) \,^{*}} \left( {\frac{\partial u_{0} }{\partial x}\,r\cos \,\theta +\frac{\partial u_{0} }{\partial y}\,r\sin \,\theta } \right) \\&\quad +2\,\sum \limits _{n=1}^\infty {\,B_{n}^{\left( {02} \right) \,^{*}} \,\left[ {\frac{\partial u_{0} }{\partial x}\,\sum \limits _{k=1}^\infty {\frac{\left( {\pi nr} \right) ^{4k-1}\cos \,\left( {4k-1} \right) \,\theta }{\left( {4k-1} \right) \,!}+\frac{\partial u_{0} }{\partial y}\,\sum \limits _{k=1}^\infty {\frac{\left( {\pi nr} \right) ^{4k-1}\sin \,\left( {4k-1} \right) \,\theta }{\left( {4k-1} \right) \,!}} } } \right] }, \end{aligned}$$
or
$$\begin{aligned}&B_{0}^{\left( {02} \right) \,^{*}} \left( {\frac{\partial u_{0} }{\partial x}\,\xi +\frac{\partial u_{0} }{\partial y}\,\eta } \right) +\sum \limits _{n=1}^\infty {\,B_{n}^{\left( {02} \right) \,^{*}} \left[ {\frac{\partial u_{0} }{\partial x}\,\left( {\sinh \,\pi n\xi \,\cos \,\pi n\eta -\cosh \,\pi n\eta \,\sin \,\pi n\xi } \right) } \right. } \nonumber \\&\quad \left. {+\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi {n}\eta \,\cos \,\uppi {n}\xi -\cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } \right] = B_{0}^{\left( {02} \right) \,^{*}} \left( {\frac{\partial u_{0} }{\partial x}\,r\cos \,\theta +\frac{\partial u_{0} }{\partial y}\,r\sin \,\theta } \right) \nonumber \\&\quad +2\,\sum \limits _{k=1}^\infty \,\frac{\pi ^{4k-1}}{\left( {4k-1} \right) \,!}\left( {\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} n^{4k-1}} } \right) \,\left( \frac{\partial u_{0} }{\partial x}r^{4k-1}\cos \,\left( {4k-1} \right) \,\theta \right. \nonumber \\&\quad \left. +\frac{\partial u_{0} }{\partial y}r^{4k-1}\sin \,\left( {4k-1} \right) \,\theta \right) . \end{aligned}$$
(3.29)
Observe that now the right-hand side of (
3.29) is convergent for all values of
\(0\le r<\infty \).
The correcting terms of the (03) approximation follow:
$$\begin{aligned}&u_{1}^{-\;\left( {03} \right) } =A_{10}^{\left( {03} \right) } \,r\cos \,\theta +A_{20}^{\left( {03} \right) } \,r\sin \,\theta \nonumber \\&\quad +\sum \limits _{k=1}^\infty {\left( {A_{1k}^{\left( {03} \right) } \,r^{4k-1}\cos \,\left( {4k-1} \right) \,\theta +A_{2k}^{\left( {03} \right) } \,r^{4k-1}\sin \,\left( {4k-1} \right) \,\theta } \right) } ; \end{aligned}$$
(3.30)
$$\begin{aligned}&u_{1}^{+\;\left( {03} \right) } =\frac{B_{10}^{\left( {03} \right) } }{r}\,\cos \,\theta +\frac{B_{20}^{\left( {03} \right) } }{r}\,\sin \,\theta + \sum \limits _{k=1}^\infty {\left( {B_{1k}^{\left( {03} \right) } \,\frac{\cos \,\left( {4k-1} \right) \,\theta }{r^{4k-1}}+B_{2k}^{\left( {03} \right) } \,\frac{\sin \,\left( {4k-1} \right) \,\theta }{r^{4k-1}}} \right) } ,\nonumber \\ \end{aligned}$$
(3.31)
where the constants
$$\begin{aligned}&A_{10}^{\left( {03} \right) } =A_{10}^{{\left( {03} \right) }{*}}\frac{\partial u_{0} }{\partial x}, \quad B_{10}^{\left( {03} \right) } =B_{10}^{{\left( {03} \right) }{*}}\frac{\partial u_{0} }{\partial x}; \\&A_{20}^{\left( {03} \right) } =A_{20}^{{\left( {03} \right) }{*}}\frac{\partial u_{0} }{\partial y}, \quad B_{20}^{\left( {03} \right) } =B_{20}^{{\left( {03} \right) }{*}}\frac{\partial u_{0} }{\partial y} \end{aligned}$$
and
$$\begin{aligned}&A_{1k}^{\left( {03.n} \right) } =A_{1k}^{{\left( {03.n} \right) }{*}}\frac{\partial u_{0} }{\partial x}, \quad B_{1k}^{\left( {03.n} \right) } =B_{1k}^{{\left( {03.n} \right) }{*}}\frac{\partial u_{0} }{\partial x}; \\&A_{2k}^{\left( {03.n} \right) } =A_{2k}^{{\left( {03.n} \right) }{*}}\frac{\partial u_{0} }{\partial y}, \quad B_{2k}^{\left( {03.n} \right) } =B_{2k}^{{\left( {03.n} \right) }{*}}\frac{\partial u_{0} }{\partial y} \end{aligned}$$
are defined through solution of either the following
$$\begin{aligned} \left\{ {\begin{array}{l} A_{m0}^{{\left( {03} \right) }{*}}\,a=\frac{B_{m0}^{{\left( {03} \right) }{*}}}{a} \\ -\frac{B_{m0}^{{\left( {03} \right) }{*}}}{a^{2}}-\lambda \,A_{m0}^{{\left( {03} \right) }{*}}=\left( {\lambda -1} \right) \,B_{0}^{{\left( {02} \right) }{*}} \\ \end{array}} \right. \end{aligned}$$
or the following
$$\begin{aligned} \left\{ {\begin{array}{l} A_{mk}^{{\left( {03.n} \right) }{*}}\,a^{4k-1}=\frac{B_{mk}^{{\left( {03.n} \right) }{*}}}{a^{4k-1}} \\ -\frac{B_{mk}^{{\left( {03.n} \right) }{*}}}{\tilde{{a}}^{4k}}-\lambda \,a^{4k-2}A_{mk}^{{\left( {03.n} \right) }{*}}=\frac{2\,\left( {\lambda -1} \right) \pi ^{4k-1}a^{4k-2}}{\left( {4k-1} \right) \,!}\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} } n^{4k-1} \\ \end{array}} \right. \end{aligned}$$
(3.32)
system of equations. Therefore, we get
$$\begin{aligned}&A_{m0}^{\left( {03} \right) \,^{*}} =-\frac{\lambda -1}{\lambda +1}\,B_{0}^{\left( {02} \right) \,^{*}} =-\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\frac{\pi a^{2}}{4},\;\;B_{m0}^{\left( {03} \right) \,^{*}} =-\frac{\lambda -1}{\lambda +1}\,a^{2}B_{0}^{\left( {02} \right) \,^{*}} =-\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\frac{\pi a^{4}}{4}, \nonumber \\&A_{mk}^{{\left( {03} \right) }{*}}=-\frac{2\,\pi ^{4k-1}}{\left( {4k-1} \right) \,!}\cdot \frac{\lambda -1}{\lambda +1}\,\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} } n^{4k-1}=-\frac{2\,\pi ^{4k-1}a^{2}}{\left( {4k-1} \right) \,!}\cdot \left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\sum \limits _{n=1}^\infty {S_{n} } n^{4k-1}, \end{aligned}$$
(3.33)
$$\begin{aligned}&B_{mk}^{{\left( {03} \right) }{*}}=-\frac{2\,\pi ^{4k-1}a^{8k-2}}{\left( {4k-1} \right) \,!}\cdot \frac{\lambda -1}{\lambda +1}\,\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} } n^{4k-1}=-\frac{2\,\pi ^{4k-1}a^{8k}}{\left( {4k-1} \right) \,!}\cdot \left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\sum \limits _{n=1}^\infty {S_{n} } n^{4k-1}. \end{aligned}$$
(3.34)
In other words, the final form of the (03) approximation governed by (
3.30), (
3.31), accounting for (
3.33), (
3.34), is written in the following form:
$$\begin{aligned}&{u}_{1}^{-\;\left( {03} \right) } =-\frac{\lambda -1}{\lambda +1}\,{B}_{0}^{\left( {02} \right) \,^{*}} {r}\,\left( {\frac{\partial {u}_{0} }{\partial {x}}\,\cos \,\theta +\frac{\partial {u}_{0} }{\partial {y}}\,\sin \,\theta } \right) \nonumber \\&\quad -\frac{\lambda -1}{\lambda +1}\,2\,\left( {\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} } n^{4k-1}} \right) \,\sum \limits _{k=1}^\infty {\,\frac{\pi ^{4k-1}\,r^{4k-1}}{\left( {4k-1} \right) \,!}\,\left( {\frac{\partial u_{0} }{\partial x}\cos \,\left( {4k-1} \right) \,\theta +\frac{\partial u_{0} }{\partial y}\sin \,\left( {4k-1} \right) \,\theta } \right) } ;\nonumber \\ \end{aligned}$$
(3.35)
$$\begin{aligned}&u_{1}^{+\;\left( {03} \right) } =-\frac{\lambda -1}{\lambda +1}\,\frac{{B}_{0}^{\left( {02} \right) \,^{*}} }{{r}}\,\left( {\frac{\partial {u}_{0} }{\partial {x}}\,\cos \,\theta +\frac{\partial {u}_{0} }{\partial {y}}\,\sin \,\theta } \right) \nonumber \\&\quad -\frac{\lambda -1}{\lambda +1}\,2\,\left( {\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} } n^{4k-1}} \right) \,\sum \limits _{k=1}^\infty \,\frac{\pi ^{4k-1}a^{8k-2}}{\left( {4k-1} \right) \,!\,r^{4k-1}}\,\left( \frac{\partial u_{0} }{\partial x}\cos \,\left( {4k-1} \right) \,\theta \right. \nonumber \\&\quad \left. +\frac{\partial u_{0} }{\partial y}\sin \,\left( {4k-1} \right) \,\theta \right) . \end{aligned}$$
(3.36)
Reversing the numbering of the series due to
k in formulas (
3.35), (
3.36) yields
$$\begin{aligned}&{u}_{1}^{-\;\left( {03} \right) } =-\frac{\lambda -1}{\lambda +1}\,{B}_{0}^{\left( {02} \right) ^{*}} \left( {\frac{\partial {u}_{0} }{\partial {x}}\,\xi +\frac{\partial {u}_{0} }{\partial {y}}\,\eta } \right) -\frac{\lambda -1}{\lambda +1}\,\sum \limits _{{n}=1}^\infty {{B}_{\mathrm{n}}^{\left( {02} \right) ^{*}} } \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\uppi {n}\xi \,\cos \,\uppi {n}\eta } \right. } \right. \nonumber \\&\quad \left. {\left. {-\cosh \,\uppi {n}\eta \,\sin \,\uppi {n}\xi } \right) +\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi {n}\eta \,\cos \,\uppi {n}\xi -\cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } \right] ; \end{aligned}$$
(3.37)
$$\begin{aligned}&u_{1}^{+\;\left( {03} \right) } =-\,\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\frac{\pi a^{2}}{4}\,\left( {\frac{\partial u_{0} }{\partial x}\,\frac{\xi }{\xi ^{2}+\eta ^{2}}+\frac{\partial u_{0} }{\partial y}\,\frac{\eta }{\xi ^{2}+\eta ^{2}}} \right) \nonumber \\&\quad -\frac{\lambda -1}{\lambda +1}\,\sum \limits _{{n}=1}^\infty {{B}_{\mathrm{n}}^{\left( {02} \right) ^{*}} } \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}-\cosh \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}\,\sin \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}} \right) } \right. \nonumber \\&\quad \left. {+\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}-\cosh \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}\,\sin \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}} \right) } \right] , \end{aligned}$$
(3.38)
i.e., we have
$$\begin{aligned}&{u}_{1}^{-\;\left( {03} \right) } =-\,\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\frac{\uppi {a}^{2}}{4}\left( {\frac{\partial {u}_{0} }{\partial {x}}\,\xi +\frac{\partial {u}_{0} }{\partial {y}}\,\eta } \right) -\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,{a}^{2}\sum \limits _{{n}=1}^\infty {{S}_{\mathrm{n}} } \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\uppi {n}\xi \,\cos \,\uppi {n}\eta } \right. } \right. \nonumber \\&\quad \left. {\left. {-\cosh \,\uppi {n}\eta \,\sin \,\uppi {n}\xi } \right) +\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi {n}\eta \,\cos \,\uppi {n}\xi -\cosh \,\uppi {n}\xi \,\sin \,\uppi {n}\eta } \right) } \right] ; \end{aligned}$$
(3.39)
$$\begin{aligned}&u_{1}^{+\;\left( {03} \right) } =-\,\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,\frac{\pi a^{2}}{4}\,\left( {\frac{\partial u_{0} }{\partial x}\,\frac{\xi }{\xi ^{2}+\eta ^{2}}+\frac{\partial u_{0} }{\partial y}\,\frac{\eta }{\xi ^{2}+\eta ^{2}}} \right) \nonumber \\&\quad -\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\,{a}^{2}\,\sum \limits _{{n}=1}^\infty {{S}_{\mathrm{n}} } \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}-\cosh \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}\,\sin \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}} \right) } \right. \nonumber \\&\quad \left. {+\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}-\cosh \,\frac{\uppi {na}^{2}\xi }{\xi ^{2}+\eta ^{2}}\,\sin \,\frac{\uppi {na}^{2}\eta }{\xi ^{2}+\eta ^{2}}} \right) } \right] . \end{aligned}$$
(3.40)
In the (04) approximation, we proceed in analogous way to remove lack of compliance of the function
\(u_{1}^{+\;\left( {03} \right) } \) of (
3.40) on the external contour of the matrix.
We define for the function
$$\begin{aligned}&{u}_{11}^{\left( {04} \right) } ={A}_{0}^{\left( {04} \right) } +{B}_{0}^{\left( {04} \right) } \,\xi +\sum \limits _{\ell =1}^\infty \left[ {\,\left( {{A}_{\ell }^{\left( {04} \right) } \,\cosh \,\uppi \ell \xi +{B}_{\ell }^{\left( {04} \right) } \,\sinh \,\uppi \ell \xi } \right) \,\cos \,\uppi \ell \eta +} \right. \\&\quad \left. {\left( {{C}_{\ell }^{\left( {02} \right) } \,\cosh \,\uppi \ell \xi + {D}_{\ell }^{\left( {02} \right) } \,\sinh \,\uppi \ell \xi } \right) \,\sin \,\uppi \ell \eta } \right] \end{aligned}$$
the following boundary conditions for the cell at
\(\xi =1\) and
\(\xi =-1\):
$$\begin{aligned}&u_{11}^{\left( {04} \right) } \,\left| {\,_{\xi =1} } \right. -u_{11}^{\left( {04} \right) } \,\left| {\,_{\xi =\,-\,1} } \right. =\frac{\partial u_{0} }{\partial x}\,\frac{\lambda -1}{\lambda +1}\,B_{0}^{\left( {02} \right) \,^{*}} \,\frac{2}{1+\eta ^{2}} \nonumber \\&\quad +\frac{\partial {u}_{0} }{\partial {x}}\,\frac{\lambda -1}{\lambda +1}\,2\,\sum \limits _{{n}=1}^\infty {{B}_{\mathrm{n}}^{\left( {02} \right) ^{*}} } \left( {\sinh \,\frac{\uppi {na}^{2}}{1+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\eta }{1+\eta ^{2}}-\cosh \,\frac{\uppi {na}^{2}\eta }{1+\eta ^{2}}\,\sin \,\frac{\uppi {na}^{2}}{1+\eta ^{2}}} \right) ; \end{aligned}$$
(3.41)
$$\begin{aligned}&\frac{\partial u_{11}^{\left( {04} \right) } }{\partial \xi }\,\,\left| {\,_{\xi =1} } \right. -\frac{\partial u_{11}^{\left( {04} \right) } }{\partial \xi }\,\,\left| {\,_{\xi =\,-\,1} } \right. =-\frac{\partial u_{0} }{\partial y}\,\frac{\lambda -1}{\lambda +1}\,B_{0}^{\left( {02} \right) \,^{*}} \,\frac{4\eta }{\left( {1+\eta ^{2}} \right) ^{2}} \nonumber \\&\quad -\frac{\partial u_{0} }{\partial y}\,\frac{\lambda -1}{\lambda +1}\,2\pi a^{2}\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) \,^{*}} n\,} \left[ {\frac{2\eta }{\left( {1+\eta ^{2}} \right) ^{2}}\cosh \,\frac{\pi na^{2}\eta }{1+\eta ^{2}}\,\cos \,\frac{\pi na^{2}}{1+\eta ^{2}}} \right. \nonumber \\&\quad -\frac{1-\eta ^{2}}{\left( {1+\eta ^{2}} \right) ^{2}}\sinh \,\frac{\pi na^{2}\eta }{1+\eta ^{2}}\,\sin \,\frac{\pi na^{2}}{1+\eta ^{2}}-\frac{1-\eta ^{2}}{\left( {1+\eta ^{2}} \right) ^{2}}\sinh \,\frac{\pi na^{2}}{1+\eta ^{2}}\,\sin \,\frac{\pi na^{2}\eta }{1+\eta ^{2}}\nonumber \\&\quad - \left. {\frac{2\eta }{\left( {1+\eta ^{2}} \right) ^{2}}\cosh \,\frac{\uppi {na}^{2}}{1+\eta ^{2}}\,\cos \,\frac{\uppi {na}^{2}\eta }{1+\eta ^{2}}} \right] . \end{aligned}$$
(3.42)
Assuming
\(a\ll 1\) and developing the functions standing on the right-hand sides of the formulas (
3.41), (
3.42) into the Fourier series, allows to define the following coefficients:
$$\begin{aligned}&A_{0}^{\left( {04} \right) } =0\,;\;\;A_{\ell }^{\left( {04} \right) } =D_{\ell }^{\left( {04} \right) } =0; \end{aligned}$$
(3.43)
$$\begin{aligned}&B_{0}^{\left( {04} \right) } =\frac{\partial u_{0} }{\partial x}\,B_{0}^{\left( {04} \right) ^{*}} \,;\;\;B_{\ell }^{\left( {04} \right) } =\frac{\partial u_{0} }{\partial x}\,B_{\ell }^{\left( {04} \right) ^{*}} \,;\;\;C_{\ell }^{\left( {04} \right) } =\frac{\partial u_{0} }{\partial y}\,C_{\ell }^{\left( {04} \right) ^{*}} , \quad \ell =0\,,\;1\,,\;2\,,\;...; \end{aligned}$$
(3.44)
$$\begin{aligned}&B_{0}^{\left( {04} \right) ^{*}} =\frac{\lambda -1}{\lambda +1}\,\frac{\pi a^{2}}{4}B_{0}^{\left( {02} \right) ^{*}} +\frac{\lambda -1}{\lambda +1}\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) ^{*}} } \sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi na^{2}} \right) ^{4m-1}}{2^{2m-1}\left( {2m-1} \right) \,\left( {4m-1} \right) \,!}} \nonumber \\&\quad =\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\frac{\pi ^{2}a^{4}}{16}\left( {1+\frac{8}{\pi }\,\sum \limits _{n=1}^\infty {S_{n} } n\sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi n} \right) ^{4m-2}a^{8m-4}}{2^{2m-2}\left( {2m-1} \right) \,\left( {4m-1} \right) \,!}} } \right) ; \end{aligned}$$
(3.45)
$$\begin{aligned}&B_{\ell }^{\left( {04} \right) ^{*}} =-C_{\ell }^{{\left( {04} \right) }^{*}}=\frac{\lambda -1}{\lambda +1}\,a^{2}B_{0}^{\left( {02} \right) ^{*}} S_{\ell } +\frac{\lambda -1}{\lambda +1}\,\sum \limits _{n=1}^\infty {B_{n}^{\left( {02} \right) ^{*}} } \sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi na^{2}} \right) ^{4m-1}}{\left( {2m-1} \right) \,\left( {4m-1} \right) \,!\left( {4m-3} \right) \,!}} \, \nonumber \\&\quad \times \left( {\left( {\pi \ell } \right) ^{4m-2}S_{\ell } +\sum \limits _{k=1}^m {\left( {-1} \right) ^{\ell +k+1}\frac{\left( {4k-3} \right) \,!}{2^{2k-2}}\,\frac{\left( {\pi \ell } \right) ^{4m-4k}}{\sinh \,\pi \ell }} } \right) \nonumber \\&\quad =\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\frac{\pi a^{4}}{4}\,\left[ {S_{\ell } +4\sum \limits _{n=1}^\infty {S_{n} } n\sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi n} \right) ^{4m-2}a^{8m-4}}{\left( {2m-1} \right) \,\left( {4m-1} \right) \,!\left( {4m-3} \right) \,!}} } \right. \nonumber \\&\quad \left. {\times \left( {\left( {\pi \ell } \right) ^{4m-2}S_{\ell } +\sum \limits _{k=1}^m {\left( {-1} \right) ^{\ell +k+1}\frac{\left( {4k-3} \right) \,!}{2^{2k-2}}\,\frac{\left( {\pi \ell } \right) ^{4m-4k}}{\sinh \,\pi \ell }} } \right) } \right] . \end{aligned}$$
(3.46)
Consequently, the (04) approximation takes the following form:
$$\begin{aligned}&u_{1}^{\left( {04} \right) } =\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\frac{\pi ^{2}a^{4}}{16}\left( {1+\frac{8}{\pi }\,\sum \limits _{n=1}^\infty {S_{n} } n\sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi n} \right) ^{4m-2}a^{8m-4}}{2^{2m-2}\left( {2m-1} \right) \,\left( {4m-1} \right) \,!}} } \right) \left( {\frac{\partial u_{0} }{\partial x}\,\xi +\frac{\partial u_{0} }{\partial y}\,\eta } \right) \nonumber \\&\quad +\left( {\frac{\lambda -1}{\lambda +1}} \right) ^{2}\frac{\pi a^{4}}{4}\,\left[ {\sum \limits _{\ell =1}^\infty \, S_{\ell } +4\sum \limits _{\ell =1}^\infty \, S_{\ell } \sum \limits _{n=1}^\infty {S_{n} } n\sum \limits _{m=1}^\infty {\,\frac{\left( {-1} \right) ^{m+1}\left( {\pi ^{2}n\ell } \right) ^{4m-2}a^{8m-4}}{\left( {2m-1} \right) \,\left( {4m-1} \right) \,!\left( {4m-3} \right) \,!}} } \right. \nonumber \\&\quad \left. {+4\sum \limits _{\ell =1}^\infty \, \frac{1}{\sinh \,\uppi \ell }\,\sum \limits _{\mathrm{n}=1}^\infty {{S}_{\mathrm{n}} } {n}\sum \limits _{\mathrm{m}=1}^\infty \,\frac{\left( {-1} \right) ^{\mathrm{m}+1}\left( {\uppi {n}} \right) ^{\mathrm{4m}-2}{a}^{\mathrm{8m}-4}}{\left( {{2m}-1} \right) \,\left( {{4m}-1} \right) \,!\left( {{4m}-3} \right) }\,! \,\sum \limits _{{k}=1}^{\mathrm{m}} \, \frac{\left( {-1} \right) ^{\ell +{k}+1}\,\left( {{4k}-3} \right) \,!\,\left( {\uppi \ell } \right) ^{\mathrm{4m-4k}}}{2^{\mathrm{2k}-2}}} \right] \nonumber \\&\quad \times \left[ {\frac{\partial {u}_{0} }{\partial {x}}\,\left( {\sinh \,\uppi \ell \xi \,\cos \,\uppi \ell \eta -\cosh \,\uppi \ell \eta \,\sin \,\uppi \ell \xi } \right) +\frac{\partial {u}_{0} }{\partial {y}}\left( {\sinh \,\uppi \ell \eta \,\cos \,\uppi \ell \xi -\cosh \,\uppi \ell \xi \,\sin \,\uppi \ell \eta } \right) } \right] .\nonumber \\ \end{aligned}$$
(3.47)