Recursion-transform method and potential formulae of the m×n cobweb and fan networks

doi:10.1088/1674-1056/26/9/090503

Abstract

In this paper, we made a new breakthrough, which proposes a new Recursion-Transform (RT) method with potential parameters to evaluate the nodal potential in arbitrary resistor networks. For the first time, we found the exact potential formulae of arbitrary m×n cobweb and fan networks by the RT method, and the potential formulae of infinite and semi-infinite networks are derived. As applications, a series of interesting corollaries of potential formulae are given by using the general formula, the equivalent resistance formula is deduced by using the potential formula, and we find a new trigonometric identity by comparing two equivalence results with different forms.

Keywords: recursion-transform method network model potential formula exact solution

Received: 14 May 2017
Revised: 07 June 2017
Accepted manuscript online:

PACS:	05.50.+q	(Lattice theory and statistics)
	84.30.Bv	(Circuit theory)
	89.20.Ff	(Computer science and technology)
	41.20.Cv	(Electrostatics; Poisson and Laplace equations, boundary-value problems)

Fund:

Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20161278).

Corresponding Authors: Zhi-Zhong Tan
E-mail: tanz@ntu.edu.cn,tanzzh@163.com

Cite this article:

Zhi-Zhong Tan(谭志中) Recursion-transform method and potential formulae of the m×n cobweb and fan networks 2017 Chin. Phys. B 26 090503

[1]	Kirchhoff G 1847 Ann. Phys. Chem. 148 497
[2]	Kirkpatrick S 1973 Rev. Mod. Phys. 45 574
[3]	Pennetta C, Alfinito E, Reggiani L, Fantini F, DeMunari I and Scorzoni A 2004 Phys. Rev. B 70 174305
[4]	Redner S 2001 A Guide to First-Passage Processes (Cambridge: Cambridge University Press)
[5]	Katsura S and Inawashiro S 1971 J. Math. Phys. 12 1622
[6]	Doyle P G and Snell J L 1984 Random Walks and Electrical Networks, The Mathematical Association of America, Washington, DC
[7]	Novak L and Gibbons A 2009 Hybrid Graph Theory and Network Analysis (Cambridge: Cambridge University Press)
[8]	Goodrich C P, Liu A J and Nagel S R 2012 Phys. Rev. Lett. 109 095704
[9]	Izmailian N S, Priezzhev V B, Philippe R and Hu C K 2005 Phys. Rev. Lett. 95 260602
[10]	Dobrosavljević V and Kotliar G 1997 Phys. Rev. Lett. 78 3943
[11]	Vollhardt D, Byczuk K and Kollar M 2011 Springer Series in Solid-State Sciences 171 203
[12]	Caracciolo R, De Pace A, Feshbach H and Molinari A 1998 Ann. Phys. 262 105
[13]	Weinan E and Wang X P 2000 SIAM J. Numer. Sci. Comp. 38 1647
[14]	Garcia-Cervera C J, Gimbutas Z and E W 2003 J. Comput. Phys. 184 37
[15]	Lai M C and Wang W C 2002 Numer. Methods Partial Differ. Equ. 18 56
[16]	Fornberg B 1996 A Practical Guide to Pseudo Spectral Methods (Cambridge: Cambridge University Press) p. 17
[17]	Houstis E, Lynch R and Rice J 1978 J. Comput. Phys. 27 323
[18]	Borges L and Daripa P 2001 J. Comput. Phys. 169 151
[19]	Kamrin K and Koval G 2012 Phys. Rev. Lett. 108 178301
[20]	Cserti J 2000 Am. J. Phys. 68 896
[21]	Giordano S 2005 Int. J. Circ. Theor. Appl. 33 519
[22]	Wu F Y 2004 J. Phys. A: Math. Gen. 37 6653
[23]	Tzeng W J and Wu F Y 2006 J. Phys.A: Math. Gen. 39 8579
[24]	Izmailian N S and Kenna R and Wu F Y 2014 J. Phys. A: Math. Theor. 47 035003
[25]	Izmailian N S and Kenna R 2014 J. Stat. Mech. E 09 1742
[26]	Tan Z Z 2015 Chin. Phys. B 24 020503
[27]	Tan Z Z 2015 Phys. Rev. E 91 052122
[28]	Tan Z Z 2015 Sci. Rep. 5 11266
[29]	Tan Z Z, Zhou L and Yang J H 2013 J. Phys. A: Math. Theor. 46 195202
[30]	Tan Z Z, Essam J W and Wu F Y 2014 Phys. Rev. E 90 012130
[31]	Essam J W, Tan Z Z and Wu F Y 2014 Phys. Rev. E 90 032130
[32]	Tan Z Z 2015 Int. J. Circ. Theor. Appl. 43 1687
[33]	Essam J W, Izmailian N S, Kenna R and Tan Z Z 2015 Roy. Soc. Open Sci. 2 140420
[34]	Tan Z Z 2016 Chin. Phys. B. 25 050504
[35]	Tan Z Z 2017 Commun. Theor. Phys. 67 280
[36]	Tan Z Z and Zhang Q H 2017 Acta Phys. Sin. 66 070501 (in Chinese)
[37]	He Z W, Zhan M, Wang J X and Yao C G 2017 Fron. Phys. 12 120701
[38]	Tyson J J, Chen K and Novak B 2001 Nat. Rev. Mol. Cell Biol. 2 908
[39]	Xia Q Z Liu L L, Ye W M and Hu G 2011 New J. Phys. 13 083002
[40]	Davidich M I and Bornholdt S 2008 PLoS ONE 3 e1672
[41]	Davidich M I and Bornholdt S 2013 PLoS ONE 8 e71786
[42]	Bornholdt S and Rohlf T 2000 Phys. Rev. Lett. 84 6114
[43]	Davidich M and Bornholdt S 2008 J. Theor. Biol. 255 269

[1]	Force-constant-decayed anisotropic network model: An improved method for predicting RNA flexibility Wei-Bu Wang(王韦卜), Xing-Yuan Li(李兴元), and Ji-Guo Su(苏计国). Chin. Phys. B, 2022, 31(6): 068704.
[2]	Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions Cui-Xian Guo(郭翠仙) and Shu Chen(陈澍). Chin. Phys. B, 2022, 31(1): 010313.
[3]	Enhancements of the Gaussian network model in describing nucleotide residue fluctuations for RNA Wen-Jing Wang(王文静) and Ji-Guo Su(苏计国). Chin. Phys. B, 2021, 30(5): 058701.
[4]	Exact scattering states in one-dimensional Hermitian and non-Hermitian potentials Ruo-Lin Chai(柴若霖), Qiong-Tao Xie(谢琼涛), Xiao-Liang Liu(刘小良). Chin. Phys. B, 2020, 29(9): 090301.
[5]	Exact solution of the (1+2)-dimensional generalized Kemmer oscillator in the cosmic string background with the magnetic field Yi Yang(杨毅), Shao-Hong Cai(蔡绍洪), Zheng-Wen Long(隆正文), Hao Chen(陈浩), Chao-Yun Long(龙超云). Chin. Phys. B, 2020, 29(7): 070302.
[6]	Exact analytical results for a two-level quantum system under a Lorentzian-shaped pulse field Qiong-Tao Xie(谢琼涛), Xiao-Liang Liu(刘小良). Chin. Phys. B, 2020, 29(6): 060305.
[7]	Unified approach to various quantum Rabi models witharbitrary parameters Xiao-Fei Dong(董晓菲), You-Fei Xie(谢幼飞), Qing-Hu Chen(陈庆虎). Chin. Phys. B, 2020, 29(2): 020302.
[8]	Dislocation neutralizing in a self-organized array of dislocation and anti-dislocation Feng-Lin Deng(邓凤麟), Xiang-Sheng Hu(胡湘生), Shao-Feng Wang(王少峰). Chin. Phys. B, 2019, 28(11): 116103.
[9]	Integrability classification and exact solutions to generalized variable-coefficient nonlinear evolution equation Han-Ze Liu(刘汉泽), Li-Xiang Zhang(张丽香). Chin. Phys. B, 2018, 27(4): 040202.
[10]	The global monopole spacetime and its topological charge Hongwei Tan(谭鸿威), Jinbo Yang(杨锦波), Jingyi Zhang(张靖仪), Tangmei He(何唐梅). Chin. Phys. B, 2018, 27(3): 030401.
[11]	Exact solutions of an Ising spin chain with a spin-1 impurity Xuchu Huang(黄旭初). Chin. Phys. B, 2017, 26(3): 037501.
[12]	Two-point resistance of an m×n resistor network with an arbitrary boundary and its application in RLC network Zhi-Zhong Tan(谭志中). Chin. Phys. B, 2016, 25(5): 050504.
[13]	Application of asymptotic iteration method to a deformed well problem Hakan Ciftci, H F Kisoglu. Chin. Phys. B, 2016, 25(3): 030201.
[14]	Bright and dark soliton solutions for some nonlinear fractional differential equations Ozkan Guner, Ahmet Bekir. Chin. Phys. B, 2016, 25(3): 030203.
[15]	Interplay between spin frustration and magnetism in the exactly solved two-leg mixed spin ladder Yan Qi(齐岩), Song-Wei Lv(吕松玮), An Du(杜安), Nai-sen Yu(于乃森). Chin. Phys. B, 2016, 25(11): 117501.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Recursion-transform method and potential formulae of the m×n cobweb and fan networks

Cite this article:

Online attention