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Chinese Physics B
Vol. 29, Issue 8, (2020)

Electrical properties of m × n cylindrical network

Zhi-Zhong Tan^1、† and Zhen Tan²

Author Affiliations

¹Department of Physics, Nantong University, Nantong 22609, China

²School of Information Science and Technology, Nantong University, Nantong 6019, China

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DOI: 10.1088/1674-1056/ab96a7 Cite this Article

Zhi-Zhong Tan, Zhen Tan. Electrical properties of m × n cylindrical network[J]. Chinese Physics B, 2020, 29(8): Copy Citation Text

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Abstract

We consider the problem of electrical properties of an m × n cylindrical network with two arbitrary boundaries, which contains multiple topological network models such as the regular cylindrical network, cobweb network, globe network, and so on. We deduce three new and concise analytical formulae of potential and equivalent resistance for the complex network of cylinders by using the RT-V method (a recursion-transform method based on node potentials). To illustrate the multiplicity of the results we give a series of special cases. Interestingly, the results obtained from the resistance formulas of cobweb network and globe network obtained are different from the results of previous studies, which indicates that our research work creates new research ideas and techniques. As a byproduct of the study, a new mathematical identity is discovered in the comparative study.

Keywords

complex boundaries cylindrical network electrical properties Laplace equation RT-V method

$$ \begin{eqnarray}\begin{array}{lll} & & {R}_{m\times n}({d}_{1},{d}_{2})\\ & = & \displaystyle \frac{{r}_{0}}{n+1}\left(|{y}_{1}-{y}_{2}|-\displaystyle \frac{{({y}_{1}-{y}_{2})}^{2}}{m}\right)\\ & & +\displaystyle \frac{r}{m}|{x}_{1}-{x}_{2}|+\displaystyle \frac{1}{m(n+1)}\displaystyle \sum _{i=1}^{m-1}\\ & & \times \displaystyle \sum _{j=1}^{n}\displaystyle \frac{{C}_{{x}_{1},j}^{2}+{C}_{{x}_{2},j}^{2}-2{C}_{{x}_{1},j}{C}_{{x}_{2},j}\cos ({y}_{2}-{y}_{1}){\theta }_{i}}{{r}_{0}^{-1}(1-\cos {\theta }_{i})+{r}^{-1}(1-\cos {\phi }_{j})},\end{array}\end{eqnarray}$$

(1)

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$$ \begin{eqnarray}\begin{array}{lll} & & {C}_{{y}_{k}-y}^{(i)}=\cos ({y}_{k}-y){\theta }_{i},{\theta }_{i}=2i\pi /m,\end{array}\end{eqnarray}$$

(2)

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$$ \begin{eqnarray}\begin{array}{lll} & & {\lambda }_{i}=h+1-h\cos {\theta }_{i}+\sqrt{{(h+1-h\cos {\theta }_{i})}^{2}-1},\\ & & {\bar{\lambda }}_{i}=h+1-h\cos {\theta }_{i}-\sqrt{{(h+1-h\cos {\theta }_{i})}^{2}-1},\end{array}\end{eqnarray}$$

(3)

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$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{k}^{(i)}=({\lambda }_{i}^{k}-{\bar{\lambda }}_{i}^{k})/({\lambda }_{i}-{\bar{\lambda }}_{i}),{\rm{\Delta }}{F}_{k}^{(i)}={F}_{k+1}^{(i)}-{F}_{k}^{(i)},\end{array}\end{eqnarray}$$

(4)

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$$ \begin{eqnarray}\begin{array}{lll} & & {\alpha }_{s,x}^{(i)}={\rm{\Delta }}{F}_{x}^{(i)}+({h}_{s}-1){\rm{\Delta }}{F}_{x-1}^{(i)},{h}_{s}={r}_{s}/{r}_{0},\end{array}\end{eqnarray}$$

(5)

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$$ \begin{eqnarray}\begin{array}{lll} & & {\beta }_{x\vee {x}_{s}}^{(i)}=\left\{\begin{array}{ll}{\beta }_{x,{x}_{s}}^{(i)}={\alpha }_{1,x}^{(i)}{\alpha }_{2,n-{x}_{s}}^{(i)}, & \,\,{\rm{if}}\,x\leqslant {x}_{s},\\ {\beta }_{{x}_{s},x}^{(i)}={\alpha }_{1,{x}_{s}}^{(i)}{\alpha }_{2,n-x}^{(i)}, & \,\,{\rm{if}}\,x\geqslant {x}_{s},\end{array}\right.\end{array}\end{eqnarray}$$

(6)

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$$ \begin{eqnarray}\begin{array}{lll} & & {G}_{n}^{(i)}={F}_{n+1}^{(i)}+({h}_{1}+{h}_{2}-2){F}_{n}^{(i)}+({h}_{1}-1)({h}_{2}-1){F}_{n-1}^{(i)}.\end{array}\end{eqnarray}$$

(7)

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$$ \begin{eqnarray}\begin{array}{ll}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}= & \displaystyle \frac{{x}_{1}-{x}_{\tau }}{m}r\\ & +\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\,\end{array}\end{eqnarray}$$

(8)

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$$ \begin{eqnarray}\begin{array}{lll}{x}_{\tau } & = & \{{x}_{1},0\leqslant x\leqslant {x}_{1}\}\cup \{x,{x}_{1}\leqslant x\leqslant {x}_{2}\}\\ & & \cup \{{x}_{2},{x}_{2}\leqslant x\leqslant n\},\end{array}\end{eqnarray}$$

(9)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}= & & \displaystyle \frac{{x}_{2}-{x}_{\tau }}{m}r\\ & & +\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(10)

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$$ \begin{eqnarray}\begin{array}{lll}{R}_{m\times n}({d}_{1},{d}_{2})= & & \displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r\\ & & +\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(11)

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$$ \begin{eqnarray}\begin{array}{lll} & & {V}_{k+1}^{(0)}=(2+2h){V}_{k}^{(0)}-{V}_{k-1}^{(0)}-h{V}_{k}^{(m-1)}-h{V}_{k}^{(1)},\\ & & {V}_{k+1}^{(i)}=(2+2h){V}_{k}^{(i)}-{V}_{k-1}^{(i)}-h{V}_{k}^{(i-1)}-h{V}_{k}^{(i+1)},\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(12)

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$$ \begin{eqnarray}{{\boldsymbol{V}}}_{k+1}={{\boldsymbol{B}}}_{m}{{\boldsymbol{V}}}_{k}-{{\boldsymbol{V}}}_{k-1}-r{{\boldsymbol{I}}}_{k}{\delta }_{k,x}({\delta }_{y,{y}_{1}}-{\delta }_{y,{y}_{2}}),\end{eqnarray}$$

(13)

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$$ \begin{eqnarray}{{\boldsymbol{V}}}_{k}={\left[{V}_{k}^{(0)},{V}_{k}^{(1)},{V}_{k}^{(2)},\ldots,{V}_{k}^{(m-1)}\right]}^{{\rm{T}}},\end{eqnarray}$$

(14)

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$$ \begin{eqnarray}{{\boldsymbol{I}}}_{k}={[J,J,J,\ldots,J]}^{{\rm{T}}},\end{eqnarray}$$

(15)

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$$ \begin{eqnarray}{{\boldsymbol{B}}}_{m}=\left(\begin{array}{ccccc}2+2h & -h & 0 & 0 & -h\\ -h & 2(1+h) & -h & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & -h & 2(1+h) & -h\\ -h & 0 & 0 & -h & 2+2h\end{array}\right).\end{eqnarray}$$

(16)

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$$ \begin{eqnarray}\begin{array}{lll} & & {h}_{1}{{\boldsymbol{V}}}_{1}=[{{\boldsymbol{B}}}_{m}-(2-{h}_{1}){\boldsymbol{E}}]{{\boldsymbol{V}}}_{0},\end{array}\end{eqnarray}$$

(17)

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$$ \begin{eqnarray}\begin{array}{lll} & & {h}_{2}{{\boldsymbol{V}}}_{n-1}=[{{\boldsymbol{B}}}_{m}-(2-{h}_{2}){\boldsymbol{E}}]{{\boldsymbol{V}}}_{n},\end{array}\end{eqnarray}$$

(18)

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$$ \begin{eqnarray}{t}_{i}=2(1+h)-2h\cos {\theta }_{i},\end{eqnarray}$$

(19)

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$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{P}}}_{m}{{\boldsymbol{B}}}_{m}={\rm{diag}}\{{t}_{0},{t}_{1},\ldots,{t}_{m-1}\}{{\boldsymbol{P}}}_{m},\end{array}\end{eqnarray}$$

(20)

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$$ \begin{eqnarray}\begin{array}{lll} & & {{\boldsymbol{X}}}_{k}={{\boldsymbol{P}}}_{m}{{\boldsymbol{V}}}_{k}\,\,\,{\rm{or}}\,\,\,{{\boldsymbol{V}}}_{k}={({{\boldsymbol{P}}}_{m})}^{-1}{{\boldsymbol{X}}}_{k},\end{array}\end{eqnarray}$$

(21)

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$$ \begin{eqnarray}{P}_{i}=[{g}_{0,i},\,{g}_{1,i},\,{g}_{2,i},\,\,\ldots,\,\,{g}_{m-1,i}],\end{eqnarray}$$

(22)

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$$ \begin{eqnarray}{g}_{k,i}=\exp ({\rm{i}}k{\theta }_{i})\,\,\,{\rm{and}}\,\,{\theta }_{i}=2i\pi /m,\,(i\geqslant 0),\end{eqnarray}$$

(23)

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$$ \begin{eqnarray}{X}_{k+1}^{(i)}={t}_{i}{X}_{k}^{(i)}-{X}_{k-1}^{(i)}-rJ({\delta }_{{x}_{1},k}{g}_{{y}_{1},i}-{\delta }_{{x}_{2},k}{g}_{{y}_{2},i}),\end{eqnarray}$$

(24)

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$$ \begin{eqnarray}\begin{array}{lll} & & {h}_{1}{X}_{1}^{(i)}=({t}_{i}+{h}_{1}-2){X}_{0}^{(i)},\end{array}\end{eqnarray}$$

(25)

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$$ \begin{eqnarray}\begin{array}{lll} & & {h}_{2}{X}_{n-1}^{(i)}=({t}_{i}+{h}_{2}-2){X}_{n}^{(i)}.\end{array}\end{eqnarray}$$

(26)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(i)}={X}_{1}^{(i)}{F}_{k}-{X}_{0}^{(i)}{F}_{k-1},\,\,\,0\leqslant k\leqslant {x}_{1},\end{array}\end{eqnarray}$$

(27)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{{x}_{1}+1}^{(i)}={t}_{i}{X}_{{x}_{1}}^{(i)}-{X}_{{x}_{1}-1}^{(i)}-rJ\exp ({\rm{i}}{y}_{1}{\theta }_{i}),\end{array}\end{eqnarray}$$

(28)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(i)}={X}_{{x}_{1}+1}^{(i)}{F}_{k-{x}_{1}}-{X}_{{x}_{1}}^{(i)}{F}_{k-{x}_{1}-1},\,\,\,{x}_{1}\leqslant k\leqslant {x}_{2},\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(29)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{{x}_{2}+1}^{(i)}={t}_{i}{X}_{{x}_{2}}^{(i)}-{X}_{{x}_{2}-1}^{(i)}+rJ\exp ({\rm{i}}{y}_{2}{\theta }_{i}),\end{array}\end{eqnarray}$$

(30)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(i)}={X}_{{x}_{2}+1}^{(i)}{F}_{k-{x}_{2}}-{X}_{{x}_{2}}^{(i)}{F}_{k-{x}_{2}-1},\,\,\,{x}_{2}\leqslant k\leqslant n,\,\,\end{array}\end{eqnarray}$$

(31)

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$$ \begin{eqnarray}{X}_{k}^{(i)}=\displaystyle \frac{{\beta }_{k\vee {x}_{1}}^{(i)}\exp ({\rm{i}}{y}_{1}{\theta }_{i})-{\beta }_{k\vee {x}_{2}}^{(i)}\exp ({\rm{i}}{y}_{2}{\theta }_{i})}{({t}_{i}-2){G}_{n}^{(i)}}rJ,\end{eqnarray}$$

(32)

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$$ \begin{eqnarray}{\lambda }_{0}={\bar{\lambda }}_{0}=1\,\,\,{\rm{and}}\,\,\,{g}_{k,0}=\exp ({\rm{i}}k{\theta }_{0})=1.\end{eqnarray}$$

(33)

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$$ \begin{eqnarray}\begin{array}{lll} & & {F}_{k}^{(0)}=k,\,\,\,{\rm{\Delta }}{F}_{k}^{(0)}=1,\\ & & {\alpha }_{s,x}^{(0)}={\rm{\Delta }}{F}_{x}^{(0)}+({h}_{s}-1){\rm{\Delta }}{F}_{x-1}^{(0)}={h}_{s}.\end{array}\end{eqnarray}$$

(34)

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$$ \begin{eqnarray}{X}_{1}^{(0)}={X}_{0}^{(0)},{X}_{n-1}^{(0)}={X}_{n}^{(0)}.\end{eqnarray}$$

(35)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}={X}_{0}^{(0)},\,\,(0\leqslant k\leqslant {x}_{1}),\end{array}\end{eqnarray}$$

(36)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{{x}_{1}+1}^{(0)}=2{X}_{{x}_{1}}^{(0)}-{X}_{{x}_{1}-1}^{(0)}-rJ,\end{array}\end{eqnarray}$$

(37)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}=(k-{x}_{1}){X}_{{x}_{1}+1}^{(0)}-(k-{x}_{1}-1){X}_{{x}_{1}}^{(0)},\,\,({x}_{1}\leqslant k\leqslant {x}_{2}),\end{array}\end{eqnarray}$$

(38)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{{x}_{2}+1}^{(0)}=2{X}_{{x}_{2}}^{(0)}-{X}_{{x}_{2}-1}^{(0)}+rJ,\,\,\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(39)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}=(k-{x}_{2}){X}_{{x}_{2}+1}^{(0)}-(k-{x}_{2}-1){X}_{{x}_{2}}^{(0)},\,\,({x}_{2}\leqslant k\leqslant n).\end{array}\end{eqnarray}$$

(40)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}={X}_{0}^{(0)},\,\,\,0\leqslant k\leqslant {x}_{1},\end{array}\end{eqnarray}$$

(41)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}={X}_{0}^{(0)}+({x}_{1}-k)rJ,\,\,\,{x}_{1}\leqslant k\leqslant {x}_{2},\end{array}\end{eqnarray}$$

(42)

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$$ \begin{eqnarray}\begin{array}{lll} & & {X}_{k}^{(0)}={X}_{0}^{(0)}+({x}_{1}-{x}_{2})rJ,\,\,\,{x}_{2}\leqslant k\leqslant n.\end{array}\end{eqnarray}$$

(43)

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$$ \begin{eqnarray}\begin{array}{lll}\left(\begin{array}{c}{X}_{k}^{(0)}\\ {X}_{k}^{(1)}\\ \vdots \\ {X}_{k}^{(s)}\end{array}\right) & = & \left(\begin{array}{ccccc}1 & 1 & 1 & \ldots & 1\\ 1 & \exp ({\rm{i}}{\theta }_{1}) & \exp ({\rm{i}}2{\theta }_{1}) & \ldots & \exp ({\rm{i}}s{\theta }_{1})\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \exp ({\rm{i}}{\theta }_{s}) & \exp ({\rm{i}}2{\theta }_{s}) & \vdots & \exp ({\rm{i}}s{\theta }_{s})\end{array}\right)\\ & & \times \left(\begin{array}{c}{V}_{k}^{(0)}\\ {V}_{k}^{(1)}\\ \vdots \\ {V}_{k}^{(s)}\end{array}\right),\end{array}\end{eqnarray}$$

(44)

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$$ \begin{eqnarray}{X}_{k}^{(0)}=\displaystyle \sum _{i=0}^{m-1}{V}_{k}^{(i)}.\end{eqnarray}$$

(45)

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$$ \begin{eqnarray}\displaystyle \sum _{i=0}^{m-1}{V}_{0}^{(i)}=0\iff {X}_{0}^{(0)}=0.\end{eqnarray}$$

(46)

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$$ \begin{eqnarray}{X}_{k}^{(0)}=({x}_{1}-{x}_{\tau })rJ,\,\,\,\,0\leqslant k\leqslant n,\end{eqnarray}$$

(47)

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$$ \begin{eqnarray}\displaystyle \sum _{i=0}^{m-1}{V}_{n}^{(i)}=0\iff {X}_{n}^{(0)}=0,\,\,{X}_{0}^{(0)}=({x}_{2}-{x}_{1})rJ.\end{eqnarray}$$

(48)

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$$ \begin{eqnarray}{X}_{k}^{(0)}=({x}_{2}-{x}_{\tau })rJ,\,\,\,0\leqslant k\leqslant n,\end{eqnarray}$$

(49)

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$$ \begin{eqnarray}\begin{array}{lll}\left(\begin{array}{c}{V}_{k}^{(0)}\\ {V}_{k}^{(1)}\\ \vdots \\ {V}_{k}^{(s)}\end{array}\right)= & & \displaystyle \frac{1}{m}\left(\begin{array}{cccc}1 & 1 & \cdots & 1\\ 1 & \exp (-{\rm{i}}{\theta }_{1}) & \cdots & \exp (-{\rm{i}}s{\theta }_{s})\\ \vdots & \vdots & \ddots & \vdots \\ 1 & \exp (-{\rm{i}}s{\theta }_{1}) & \cdots & \exp (-{\rm{i}}s{\theta }_{s})\end{array}\right)\\ & & \times \left(\begin{array}{c}{X}_{k}^{(0)}\\ {X}_{k}^{(1)}\\ \vdots \\ {X}_{k}^{(s)}\end{array}\right),\end{array}\end{eqnarray}$$

(50)

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$$ \begin{eqnarray}{V}_{k}^{(y)}=\displaystyle \frac{1}{m}\left({X}_{k}^{(0)}+\displaystyle \sum _{i=1}^{m-1}{X}_{k}^{(i)}\exp (-{\rm{i}}y{\theta }_{i})\right).\end{eqnarray}$$

(51)

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$$ \begin{eqnarray}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{x}_{1}-{x}_{\tau }}{m}r+\displaystyle \frac{{r}_{0}}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}+{\rm{i}}\displaystyle \frac{{r}_{0}}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}\sin [({y}_{1}-y){\theta }_{i}]-{\beta }_{{x}_{2}\vee x}^{(i)}\sin [({y}_{2}-y){\theta }_{i}]}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{eqnarray}$$

(52)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=[U({x}_{1},{y}_{1})-U({x}_{2},{y}_{2})]\displaystyle \frac{1}{J}.\end{eqnarray}$$

(53)

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$$ \begin{eqnarray}\displaystyle \frac{{U}_{m\times n}({x}_{1},{y}_{1})}{J}=\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1},{x}_{1}}^{(i)}-{\beta }_{{x}_{1},{x}_{2}}^{(i)}\cos ({y}_{2}-{y}_{1}){\theta }_{i}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}},\end{eqnarray}$$

(54)

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$$ \begin{eqnarray}\displaystyle \frac{{U}_{m\times n}({x}_{2},{y}_{2})}{J}=\displaystyle \frac{{x}_{1}-{x}_{2}}{m}r+\displaystyle \frac{{r}_{0}}{2m}\times \displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1},{x}_{2}}^{(i)}\cos ({y}_{2}-{y}_{1}){\theta }_{i}-{\beta }_{{x}_{2},{x}_{2}}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{eqnarray}$$

(55)

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$$ \begin{eqnarray}\displaystyle \frac{{U}_{m\times n}({x}_{1},{y}_{1})}{J}-\displaystyle \frac{{U}_{m\times n}({x}_{2},{y}_{2})}{J}=\displaystyle \frac{{x}_{2}-{x}_{1}}{m}r+\displaystyle \frac{{r}_{0}}{m}\displaystyle \sum _{i=1}^{m-1}\left(\displaystyle \frac{{\beta }_{{x}_{1},{x}_{1}}^{(i)}-{\beta }_{{x}_{1},{x}_{2}}^{(i)}\cos ({y}_{2}-{y}_{1}){\theta }_{i}}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}-\displaystyle \frac{{\beta }_{{x}_{1},{x}_{2}}^{(i)}\cos ({y}_{2}-{y}_{1}){\theta }_{i}-{\beta }_{{x}_{2},{x}_{2}}^{(i)}}{2(1-\cos {\theta }_{i}){G}_{n}^{(i)}}\right).\end{eqnarray}$$

(56)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{{U}_{m\times n}({x}_{1},{y}_{1})}{J}-\displaystyle \frac{{U}_{m\times n}({x}_{2},{y}_{2})}{J}=\displaystyle \frac{{x}_{2}-{x}_{1}}{m}r+\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{\beta }_{{x}_{1},{x}_{1}}^{(i)}-2{\beta }_{{x}_{1},{x}_{2}}^{(i)}\cos ({y}_{2}-{y}_{1}){\theta }_{i}+{\beta }_{{x}_{2},{x}_{2}}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{eqnarray}$$

(57)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{{x}_{1}-{x}_{\tau }}{m}r+\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\,\,\,\,\end{eqnarray}$$

(58)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{C}_{{y}_{1}-y}^{(i)}-{C}_{{y}_{2}-y}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}{\beta }_{{x}_{1}\vee x}^{(i)},\end{array}\end{eqnarray}$$

(59)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{m\times \infty }(x,y)}{J}=\displaystyle \frac{r}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{C}_{{y}_{1}-y}^{(i)}-{C}_{{y}_{2}-y}^{(i)}}{\sqrt{{(1+h-h\cos {\theta }_{i})}^{2}-1}}{\bar{\lambda }}_{i}^{|{x}_{1}-x|}.\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(60)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{{x}_{1}-{x}_{\tau }}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{1}-1){\rm{\Delta }}{F}_{n-1}^{(i)}},\end{eqnarray}$$

(61)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=-\displaystyle \frac{x}{m}r+\displaystyle \frac{{r}_{1}h}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{F}_{n-x}^{(i)}\cos ({y}_{1}-y){\theta }_{i}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{1}-1){\rm{\Delta }}{F}_{n-1}^{(i)}}.\end{eqnarray}$$

(62)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{{x}_{2}-{x}_{\tau }}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{1}-1){\rm{\Delta }}{F}_{n-1}^{(i)}},\end{eqnarray}$$

(63)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{n-x}{m}r+\displaystyle \frac{{r}_{1}h}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{F}_{n-x}^{(i)}\cos ({y}_{1}-y){\theta }_{i}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{1}-1){\rm{\Delta }}{F}_{n-1}^{(i)}}.\,\,\,\,\,\end{eqnarray}$$

(64)

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$$ \begin{eqnarray}\displaystyle \frac{{U}_{m\times n}(x,y)}{J}=\displaystyle \frac{{x}_{1}-{x}_{\tau }}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(i)}{C}_{{y}_{1}-y}^{(i)}-{\beta }_{{x}_{2}\vee x}^{(i)}{C}_{{y}_{2}-y}^{(i)}}{{F}_{n}^{(i)}},\,\,\,\,\end{eqnarray}$$

(65)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=-\displaystyle \frac{x}{m}r.\end{eqnarray}$$

(66)

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$$ \begin{eqnarray}\displaystyle \frac{U(x,y)}{J}=\displaystyle \frac{x}{m}r.\end{eqnarray}$$

(67)

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$$ \begin{eqnarray}\begin{array}{lll} & & {\lambda }_{1}={\lambda }_{3}=1+h+\sqrt{{(1+h)}^{2}-1},\\ & & {\lambda }_{2}=1+2h+\sqrt{{(1+2h)}^{2}-1}.\end{array}\end{eqnarray}$$

(68)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J} & = & \displaystyle \frac{{x}_{1}-{x}_{\tau }}{4}r+{r}_{0}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(1)}{C}_{y}^{(1)}-{\beta }_{{x}_{2}\vee x}^{(1)}{C}_{{y}_{2}-y}^{(1)}}{4{G}_{n}^{(1)}}\\ & & +\,{r}_{0}\displaystyle \frac{{\beta }_{{x}_{1}\vee x}^{(2)}{C}_{y}^{(2)}-{\beta }_{{x}_{2}\vee x}^{(2)}{C}_{{y}_{2}-y}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(69)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J}= & & {r}_{1}\displaystyle \frac{{C}_{y}^{(1)}-{C}_{1-y}^{(1)}}{4{G}_{n}^{(1)}}{\alpha }_{2,n-x}^{(1)}\\ & & +\,{r}_{1}\displaystyle \frac{{C}_{y}^{(2)}-{C}_{1-y}^{(2)}}{16{G}_{n}^{(2)}}{\alpha }_{2,n-x}^{(2)},\end{array}\end{eqnarray}$$

(70)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({A}_{x})}{J}={r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(2)}}{8{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(71)

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$$ \begin{eqnarray}\begin{array}{lll} & & {U}_{\square \times n}({B}_{x})=-{U}_{\square \times n}({A}_{x}),\end{array}\end{eqnarray}$$

(72)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({C}_{x})}{J}=-{r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(2)}}{8{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(73)

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$$ \begin{eqnarray}\begin{array}{lll} & & {U}_{\square \times n}({D}_{x})=-{U}_{\square \times n}({C}_{x}).\end{array}\end{eqnarray}$$

(74)

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$$ \begin{eqnarray}\begin{array}{ll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J}= & {r}_{1}\displaystyle \frac{{C}_{y}^{(1)}-{C}_{2-y}^{(1)}}{4{G}_{n}^{(1)}}{\alpha }_{2,n-x}^{(1)}\\ & +{r}_{1}\displaystyle \frac{{C}_{y}^{(2)}-{C}_{2-y}^{(2)}}{16{G}_{n}^{(2)}}{\alpha }_{2,n-x}^{(2)},\end{array}\end{eqnarray}$$

(75)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({A}_{x})}{J}={r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(1)}}{2{G}_{n}^{(1)}},\end{array}\end{eqnarray}$$

(76)

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$$ \begin{eqnarray}\begin{array}{lll} & & {U}_{\square \times n}({B}_{x})={U}_{\square \times n}({D}_{x})=0,\end{array}\end{eqnarray}$$

(77)

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$$ \begin{eqnarray}\begin{array}{lll} & & {U}_{\square \times n}({C}_{x})=-{U}_{\square \times n}({A}_{x}).\end{array}\end{eqnarray}$$

(78)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J} & = & -\displaystyle \frac{x}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}-{h}_{2}{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}{C}_{y}^{(1)}\\ & & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}}{C}_{y}^{(2)},\end{array}\end{eqnarray}$$

(79)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}({A}_{x})}{J}= & -\displaystyle \frac{x}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}-{h}_{2}{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}} & \\ & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}}, & \end{array}\end{eqnarray}$$

(80)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({B}_{x})}{J}=\displaystyle \frac{{U}_{\square \times n}({D}_{x})}{J}=-\displaystyle \frac{x}{4}r-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(81)

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$$ \begin{eqnarray}\begin{array}{ll}\displaystyle \frac{{U}_{\square \times n}({C}_{x})}{J}= & -\displaystyle \frac{x}{4}r-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}-{h}_{2}{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}\\ & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}}.\end{array}\end{eqnarray}$$

(82)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J} & = & -\displaystyle \frac{x}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}{C}_{y}^{(1)}-{h}_{2}{\alpha }_{1,x}^{(1)}{C}_{1-y}^{(1)}}{4{G}_{n}^{(1)}}\\ & & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}{C}_{y}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}{C}_{1-y}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(83)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({A}_{x})}{J}=-\displaystyle \frac{x}{4}r+{r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}+{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(84)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({B}_{x})}{J}=-\displaystyle \frac{x}{4}r-{r}_{2}\displaystyle \frac{{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}+{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(85)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({C}_{x})}{J}=-\displaystyle \frac{x}{4}r-{r}_{1}\displaystyle \frac{{\alpha }_{2,n-x}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}+{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(86)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({D}_{x})}{J}=-\displaystyle \frac{x}{4}r+{r}_{2}\displaystyle \frac{{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}+{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}}.\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(87)

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$$ \begin{eqnarray}\begin{array}{lll}\displaystyle \frac{{U}_{\square \times n}(x,y)}{J} & = & -\displaystyle \frac{x}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}{C}_{y}^{(1)}-{h}_{2}{\alpha }_{1,x}^{(1)}{C}_{2-y}^{(1)}}{4{G}_{n}^{(1)}}\\ & & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}{C}_{y}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}{C}_{2-y}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(88)

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$$ \begin{eqnarray}\begin{array}{ll}\displaystyle \frac{{U}_{\square \times n}({A}_{x})}{J}= & -\displaystyle \frac{x}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}+{h}_{2}{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}\\ & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\end{array}\end{eqnarray}$$

(89)

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$$ \begin{eqnarray}\begin{array}{lll} & & \displaystyle \frac{{U}_{\square \times n}({B}_{x})}{J}=\displaystyle \frac{{U}_{\square \times n}({D}_{x})}{J}=-\displaystyle \frac{x}{4}r-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}},\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(90)

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$$ \begin{eqnarray}\begin{array}{ll}\displaystyle \frac{{U}_{\square \times n}({C}_{x})}{J}= & -\displaystyle \frac{x}{4}r-{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(1)}+{h}_{2}{\alpha }_{1,x}^{(1)}}{4{G}_{n}^{(1)}}\\ & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n-x}^{(2)}-{h}_{2}{\alpha }_{1,x}^{(2)}}{16{G}_{n}^{(2)}}.\end{array}\end{eqnarray}$$

(91)

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$$ \begin{eqnarray}\begin{array}{ll}{R}_{m\times n}({d}_{1},{d}_{2})= & \displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r\\ & +\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{array}\end{eqnarray}$$

(92)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r+\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i})[{F}_{n+1}^{(i)}+({h}_{2}-1){F}_{n}^{(i)}]},\,\,\,\,\,\,\,\,\end{eqnarray}$$

(93)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}+({h}_{1}-1){\rm{\Delta }}{F}_{n-1}^{(i)}},\end{eqnarray}$$

(94)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}},\end{eqnarray}$$

(95)

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$$ \begin{eqnarray}{R}_{m\times n}({d}_{1},{d}_{2})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{F}_{{x}_{1}}^{(i)}{F}_{n-{x}_{1}}^{(i)}-2{F}_{{x}_{1}}^{(i)}{F}_{n-{x}_{2}}^{(i)}\cos (y{\theta }_{i})+{F}_{{x}_{2}}^{(i)}{F}_{n-{x}_{2}}^{(i)}}{{F}_{n}^{(i)}},\,\,\,\,\,\,\,\,\end{eqnarray}$$

(96)

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$$ \begin{eqnarray}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m}r+\displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(i)}+{h}_{2}{\alpha }_{1,n}^{(i)}-2{h}_{1}{h}_{2}\cos (y{\theta }_{i})}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\end{eqnarray}$$

(97)

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$$ \begin{eqnarray}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m}r+\displaystyle \frac{{r}_{0}}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(i)}-\cos (y{\theta }_{i})}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}.\end{eqnarray}$$

(98)

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$$ \begin{eqnarray}{R}_{m\times n}(\{0,0\},\{n,y\})=\displaystyle \frac{n}{m}r+\displaystyle \frac{r}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{F}_{n}^{(i)}}{{\rm{\Delta }}{F}_{n}^{(i)}}.\end{eqnarray}$$

(99)

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$$ \begin{eqnarray}\begin{array}{lll}{R}_{m\times n}(\{{x}_{1},0\},\{{x}_{2},0\})= \displaystyle \frac{|{x}_{2}-{x}_{1}|}{m}r\\ + \displaystyle \frac{{r}_{0}}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){G}_{n}^{(i)}}.\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(100)

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$$ \begin{eqnarray}{\lambda }_{1}={\lambda }_{2}=1+\displaystyle \frac{3}{2}h+\sqrt{{\left(1+\displaystyle \frac{3}{2}h\right)}^{2}-1}.\end{eqnarray}$$

(101)

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$$ \begin{eqnarray}{R}_{{\rm{\Delta }}\times n}({A}_{{x}_{1}},{P}_{k})=\displaystyle \frac{|k-{x}_{1}|}{3}r+\displaystyle \frac{2{r}_{0}}{9}\left(\displaystyle \frac{{\beta }_{1,1}^{(1)}-2{\beta }_{1,2}^{(1)}\cos (2\pi y/3)+{\beta }_{2,2}^{(1)}}{{G}_{n}^{(1)}}\right),\,\end{eqnarray}$$

(102)

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$$ \begin{eqnarray}{R}_{{\rm{\Delta }}\times n}({A}_{{x}_{1}},{A}_{{x}_{2}})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{3}r+\displaystyle \frac{2{r}_{0}}{9}\left(\displaystyle \frac{{\beta }_{{x}_{1},{x}_{1}}^{(1)}-2{\beta }_{{x}_{1},{x}_{2}}^{(1)}+{\beta }_{{x}_{2},{x}_{2}}^{(1)}}{{G}_{n}^{(1)}}\right),\end{eqnarray}$$

(103)

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$$ \begin{eqnarray}\begin{array}{lll} & & {R}_{{\rm{\Delta }}\times n}({A}_{0},{A}_{x})=\displaystyle \frac{x}{3}r+\displaystyle \frac{2{r}_{0}}{9}\left(\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}-2{\rm{\Delta }}{F}_{n-x}^{(1)}+{\rm{\Delta }}{F}_{x}^{(1)}{\rm{\Delta }}{F}_{n-x}^{(1)}}{{F}_{n+1}^{(1)}}\right),\end{array}\end{eqnarray}$$

(104)

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$$ \begin{eqnarray}\begin{array}{lll} & & {R}_{{\rm{\Delta }}\times n}({A}_{0},{A}_{n})=\displaystyle \frac{n}{3}r+\displaystyle \frac{4{r}_{0}}{9}\left(\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}-1}{{F}_{n+1}^{(1)}}\right).\end{array}\end{eqnarray}$$

(105)

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$$ \begin{eqnarray}{R}_{{\rm{\Delta }}\times n}({A}_{{x}_{1}},{B}_{k})={R}_{{\rm{\Delta }}\times n}({A}_{{x}_{1}},{C}_{k})=\displaystyle \frac{|k-{x}_{1}|}{3}r+\displaystyle \frac{2{r}_{0}}{9}\left(\displaystyle \frac{{\beta }_{1,1}^{(1)}+{\beta }_{1,2}^{(1)}+{\beta }_{2,2}^{(1)}}{{G}_{n}^{(1)}}\right).\end{eqnarray}$$

(106)

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$$ \begin{eqnarray}\begin{array}{lll} & & {R}_{{\rm{\Delta }}\times n}({A}_{0},{B}_{k})={R}_{{\rm{\Delta }}\times n}({A}_{0},{C}_{k})=\displaystyle \frac{k}{3}r+\displaystyle \frac{2{r}_{0}}{9}\left(\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}+{\rm{\Delta }}{F}_{n-k}^{(1)}+{\rm{\Delta }}{F}_{k}^{(1)}{\rm{\Delta }}{F}_{n-k}^{(1)}}{{F}_{n+1}^{(1)}}\right),\end{array}\end{eqnarray}$$

(107)

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$$ \begin{eqnarray}\begin{array}{lll} & & {R}_{{\rm{\Delta }}\times n}({A}_{0},{B}_{n})={R}_{{\rm{\Delta }}\times n}({A}_{0},{C}_{n})=\displaystyle \frac{n}{3}r+2{r}_{0}\displaystyle \frac{2{\rm{\Delta }}{F}_{n}^{(1)}+1}{9{F}_{n+1}^{(1)}}.\end{array}\end{eqnarray}$$

(108)

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$$ \begin{eqnarray}{R}_{{\rm{\Delta }}\times n}({A}_{k},{B}_{k})={R}_{{\rm{\Delta }}\times n}({A}_{k},{C}_{k})=2{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{k}^{(1)}{\rm{\Delta }}{F}_{n-k}^{(1)}}{3{F}_{n+1}^{(1)}}.\end{eqnarray}$$

(109)

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$$ \begin{eqnarray}\begin{array}{lll} & & {\lambda }_{1}={\lambda }_{3}=1+h+\sqrt{{h}^{2}+2h},\\ & & {\lambda }_{2}=1+2h+2\sqrt{{h}^{2}+h}.\end{array}\end{eqnarray}$$

(110)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{{x}_{1}},{P}_{{x}_{2}})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{4}r+{r}_{0}\displaystyle \frac{{\beta }_{1,1}^{(1)}-2{\beta }_{1,2}^{(1)}\cos (y\pi /2)+{\beta }_{2,2}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{\beta }_{1,1}^{(2)}-2{\beta }_{1,2}^{(2)}\cos (y\pi )+{\beta }_{2,2}^{(2)}}{16{G}_{n}^{(2)}},\end{eqnarray}$$

(111)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{P}_{k})=\displaystyle \frac{k}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{}-2{h}_{1}{\alpha }_{2,n-k}^{(1)}\cos (y\pi /2)+{\alpha }_{1,k}^{(1)}{\alpha }_{2,n-k}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(2)}-2{h}_{1}{\alpha }_{2,n-k}^{(2)}\cos (y\pi )+{\alpha }_{1,k}^{(2)}{\alpha }_{2,n-k}^{(2)}}{16{G}_{n}^{(2)}},\end{eqnarray}$$

(112)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{P}_{n})=\displaystyle \frac{n}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{}-2{h}_{1}{h}_{2}\cos (y\pi /2)+{h}_{2}{\alpha }_{1,n}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(2)}-2{h}_{1}{h}_{2}\cos (y\pi )+{h}_{2}{\alpha }_{1,n}^{(2)}}{16{G}_{n}^{(2)}}.\end{eqnarray}$$

(113)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{{x}_{1}},{P}_{{x}_{1}})={r}_{0}\displaystyle \frac{{\beta }_{1,1}^{(1)}[1-\cos (y\pi /2)]}{2{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{\beta }_{1,1}^{(2)}[1-\cos (y\pi )]}{8{G}_{n}^{(2)}},\end{eqnarray}$$

(114)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{{x}_{1}},{P}_{{x}_{1}})={r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{{x}_{1}}^{(1)}{\rm{\Delta }}{F}_{n-{x}_{1}}^{(1)}[1-\cos (y\pi /2)]}{2{F}_{n+1}^{(1)}}+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{{x}_{1}}^{(2)}{\rm{\Delta }}{F}_{n-{x}_{1}}^{(2)}[1-\cos (y\pi )]}{8{F}_{n+1}^{(2)}}.\end{eqnarray}$$

(115)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{A}_{k})=\displaystyle \frac{k}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(1)}-2{h}_{1}{\alpha }_{2,n-k}^{(1)}+{\alpha }_{1,k}^{(1)}{\alpha }_{2,n-k}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(2)}-2{h}_{1}{\alpha }_{2,n-k}^{(2)}+{\alpha }_{1,k}^{(2)}{\alpha }_{2,n-k}^{(2)}}{16{G}_{n}^{(2)}}.\end{eqnarray}$$

(116)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{A}_{n})=\displaystyle \frac{n}{4}r+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}-1}{2{F}_{n+1}^{(1)}}+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(2)}-1}{8{F}_{n+1}^{(2)}},\end{eqnarray}$$

(117)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{B}_{k})=\displaystyle \frac{k}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(1)}+{\alpha }_{1,k}^{(1)}{\alpha }_{2,n-k}^{(1)}}{4{G}_{n}^{(1)}}+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(2)}+2{h}_{1}{\alpha }_{2,n-k}^{(2)}+{\alpha }_{1,k}^{(2)}{\alpha }_{2,n-k}^{(2)}}{16{G}_{n}^{(2)}},\end{eqnarray}$$

(118)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{B}_{n})=\displaystyle \frac{n}{4}r+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}}{2{F}_{n+1}^{(1)}}+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(2)}+1}{8{F}_{n+1}^{(2)}}.\end{eqnarray}$$

(119)

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$$ \begin{eqnarray}\begin{array}{lll}{R}_{\square \times n}({A}_{0},{C}_{k}) & = & \displaystyle \frac{k}{4}r+{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(1)}+2{h}_{1}{\alpha }_{2,n-k}^{(1)}+{\alpha }_{1,k}^{(1)}{\alpha }_{2,n-k}^{(1)}}{4{G}_{n}^{(1)}}\\ & & +{r}_{0}\displaystyle \frac{{h}_{1}{\alpha }_{2,n}^{(2)}-2{h}_{1}{\alpha }_{2,n-k}^{(2)}+{\alpha }_{1,k}^{(2)}{\alpha }_{2,n-k}^{(2)}}{16{G}_{n}^{(2)}}.\,\,\,\,\,\,\,\,\,\end{array}\end{eqnarray}$$

(120)

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$$ \begin{eqnarray}{R}_{\square \times n}({A}_{0},{C}_{n})=\displaystyle \frac{n}{4}r+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(1)}+1}{2{F}_{n+1}^{(1)}}+{r}_{0}\displaystyle \frac{{\rm{\Delta }}{F}_{n}^{(2)}-1}{8{F}_{n+1}^{(2)}}.\end{eqnarray}$$

(121)

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$$ \begin{eqnarray}R({d}_{1},{d}_{2})=\displaystyle \frac{|{x}_{2}-{x}_{1}|}{4}r+\displaystyle \frac{{\beta }_{1,1}^{(1)}-2{\beta }_{1,2}^{(1)}\cos (y\pi /2)+{\beta }_{2,2}^{(1)}}{2{F}_{3}^{(1)}}r+\displaystyle \frac{{\beta }_{1,1}^{(2)}-2{\beta }_{1,2}^{(2)}\cos (y\pi )+{\beta }_{2,2}^{(2)}}{4{F}_{3}^{(2)}}r,\end{eqnarray}$$

(122)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({A}_{0},{A}_{3})=\displaystyle \frac{3}{4}r,\end{array}\end{eqnarray}$$

(123)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({A}_{0},{B}_{k})=\displaystyle \frac{1}{4}r+\displaystyle \frac{(h+1)r}{(2h+1)(2h+3)}+\displaystyle \frac{(2h+1)r}{2(4h+1)(4h+3)},\,\,\end{array}\end{eqnarray}$$

(124)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({A}_{0},{C}_{k})=\displaystyle \frac{1}{2}r+\displaystyle \frac{(h+1)r}{(2h+1)(2h+3)}+\displaystyle \frac{(2h+1)r}{2(4h+1)(4h+3)},\end{array}\end{eqnarray}$$

(125)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({B}_{0},{B}_{1})=R({B}_{0},{B}_{3})=\displaystyle \frac{(h+1)r}{(2h+1)(2h+3)}+\displaystyle \frac{(2h+1)r}{(4h+1)(4h+3)},\end{array}\end{eqnarray}$$

(126)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({B}_{0},{B}_{2})=\displaystyle \frac{2(h+1)r}{(2h+1)(2h+3)},\end{array}\end{eqnarray}$$

(127)

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$$ \begin{eqnarray}\begin{array}{lll} & & R({B}_{0},{C}_{k})=\displaystyle \frac{1}{4}r+\displaystyle \frac{2(h+1)-\cos (k\pi /2)}{(2h+1)(2h+3)}r+\displaystyle \frac{2(2h+1)-\cos (k\pi )}{2(4h+1)(4h+3)}r,\end{array}\end{eqnarray}$$

(128)

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$$ \begin{eqnarray}\displaystyle \frac{1}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \sum _{j=1}^{n}\displaystyle \frac{{C}_{{x}_{1},j}^{2}+{C}_{{x}_{2},j}^{2}-2{C}_{{x}_{1},j}{C}_{{x}_{2},j}\cos (y{\theta }_{i})}{(1-\cos {\theta }_{i})+{h}^{-1}(1-\cos {\phi }_{j})}=\left(\displaystyle \frac{{y}^{2}}{m}-y\right)+\displaystyle \frac{n+1}{2m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}\cos (y{\theta }_{i})+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}},\end{eqnarray}$$

(129)

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$$ \begin{eqnarray}{\lambda }_{i}=1+h-h\cos {\theta }_{i}+\sqrt{{(1+h-h\cos {\theta }_{i})}^{2}-1},\,{\bar{\lambda }}_{i}=1+h-h\cos {\theta }_{i}-\sqrt{{(1+h-h\cos {\theta }_{i})}^{2}-1}.\end{eqnarray}$$

(130)

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$$ \begin{eqnarray}\displaystyle \frac{2}{n+1}\displaystyle \sum _{i=1}^{m-1}\displaystyle \sum _{j=1}^{n}\displaystyle \frac{{[\cos ({x}_{1}+1/2){\phi }_{j}-\cos ({x}_{2}+1/2){\phi }_{j}]}^{2}}{(1-\cos {\theta }_{i})+{h}^{-1}(1-\cos {\phi }_{j})}=\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\beta }_{1,1}^{(i)}-2{\beta }_{1,2}^{(i)}+{\beta }_{2,2}^{(i)}}{(1-\cos {\theta }_{i}){F}_{n+1}^{(i)}}.\end{eqnarray}$$

(131)

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$$ \begin{eqnarray}\displaystyle \frac{1}{n+1}\displaystyle \sum _{j=1}^{n}\displaystyle \frac{{[\cos ({x}_{1}+1/2){\phi }_{j}-\cos ({x}_{2}+1/2){\phi }_{j}]}^{2}}{2+{h}^{-1}(1-\cos {\phi }_{j})}=\displaystyle \frac{{\beta }_{1,1}^{(1)}-2{\beta }_{1,2}^{(1)}+{\beta }_{2,2}^{(1)}}{4{F}_{n+1}^{(1)}},\end{eqnarray}$$

(132)

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$$ \begin{eqnarray}{\lambda }_{1}=1+2h+2\sqrt{h(1+h)},\,{\bar{\lambda }}_{1}=1+2h-2\sqrt{h(1+h)}.\end{eqnarray}$$

(133)

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$$ \begin{eqnarray}\displaystyle \frac{2}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \sum _{j=1}^{n}\displaystyle \frac{{\cos }^{2}[(x+1/2){\phi }_{j}](1-\cos y{\theta }_{i})}{(1-\cos {\theta }_{i})+{h}^{-1}(1-\cos {\phi }_{j})}=\left(\displaystyle \frac{{y}^{2}}{m}-y\right)+\displaystyle \frac{n+1}{m}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{{\rm{\Delta }}{F}_{x}^{(i)}{\rm{\Delta }}{F}_{n-x}^{(i)}}{{F}_{n+1}^{(i)}}\left(\displaystyle \frac{1-\cos (y{\theta }_{i})}{1-\cos {\theta }_{i}}\right),\end{eqnarray}$$

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$$ \begin{eqnarray}\displaystyle \frac{1}{n+1}\displaystyle \sum _{j=1}^{n}\displaystyle \frac{{({C}_{{x}_{1},j}+{C}_{{x}_{2},j})}^{2}}{2+{h}^{-1}(1-\cos {\phi }_{j})}=\displaystyle \frac{{\beta }_{1,1}^{(1)}+2{\beta }_{1,2}^{(1)}+{\beta }_{2,2}^{(1)}}{4{F}_{n+1}^{(1)}}-\displaystyle \frac{1}{n+1},\end{eqnarray}$$

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$$ \begin{eqnarray}\displaystyle \sum _{i=1}^{m-1}\displaystyle \frac{1-\cos (yi\pi /m)}{1-\cos (i\pi /m)}=y(m-y).\end{eqnarray}$$

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Zhi-Zhong Tan, Zhen Tan. Electrical properties of m × n cylindrical network[J]. Chinese Physics B, 2020, 29(8):

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