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June  2014, 9(2): 197-216. doi: 10.3934/nhm.2014.9.197

Asymptotic behaviour of flows on reducible networks

Jacek Banasiak 1, and  Proscovia Namayanja 2,

1. 

School of Mathematical Sciences, UKZN, Durban
2. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa

Received  March 2013 Revised  December 2013 Published  July 2014

In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the problem with generalized Kirchhoff's conditions, which allow for amplification and/or reduction of the flow at the nodes, on such reducible networks with sources but show that the problem becomes ill-posed if the network has a sink. Furthermore, we extend the existing results on the asymptotic periodicity of the flow to such networks. In particular, in contrast to previous papers, we consider networks with acyclic parts and we prove that such parts of the network become depleted in a finite time, an estimate of which is also provided. Finally, we show how to apply these results to open networks where a portion of the flowing material is allowed to leave the network.
Keywords: semigroups of operators, asymptotic periodicity., topological sorting algorithm, network, Transport equation, resolvent positive operators, diagraph.
Mathematics Subject Classification: Primary: 47D06, 35F45; Secondary: 05C90, 82C7.
Citation: Jacek Banasiak, Proscovia Namayanja. Asymptotic behaviour of flows on reducible networks. Networks & Heterogeneous Media, 2014, 9 (2) : 197-216. doi: 10.3934/nhm.2014.9.197
References:
[1]

W. J. Anderson, Continuous-Time Markov Chains. An Application Oriented Approach, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349. doi: 10.1112/plms/s3-54.2.321.  Google Scholar

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications, Springer Verlag, London, 2006.  Google Scholar

[4]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.  Google Scholar

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Inc., Englewood Cliffs, 1974.  Google Scholar

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B. Dorn, Flows in Infinite Networks - A Semigroup Aproach, Ph.D thesis, University of Tübingen, 2008. Google Scholar

[8]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.  Google Scholar

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.  Google Scholar

[10]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.  Google Scholar

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000.  Google Scholar

[12]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc. New York, 1959.  Google Scholar

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[14]

F.M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. & Optimization, 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6.  Google Scholar

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.  Google Scholar

[16]

T. Matrai and E. Sikolya, Asymptotic behaviour of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.  Google Scholar

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.  Google Scholar

[18]

H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.  Google Scholar

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators, Springer Verlag, Berlin, 1986.  Google Scholar

[20]

P. Namayanja, Transport on Network Structures, Ph.D thesis, UKZN, 2012. Google Scholar

[21]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[22]

E. Sikolya, Semigroups for Flows in Networks, Ph.D dissertation, University of Tübingen, 2004. Google Scholar

[23]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.  Google Scholar

show all references

References:
[1]

W. J. Anderson, Continuous-Time Markov Chains. An Application Oriented Approach, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0.  Google Scholar

[2]

W. Arendt, Resolvent positive operators, Proc. Lond. Math. Soc., 54 (1987), 321-349. doi: 10.1112/plms/s3-54.2.321.  Google Scholar

[3]

J. Banasiak and L. Arlotti, Perturbation of Positive Semigroups with Applications, Springer Verlag, London, 2006.  Google Scholar

[4]

J. Banasiak and P. Namayanja, Relative entropy and discrete Poincaré inequalities for reducible matrices, Appl. Math. Lett., 25 (2012), 2193-2197. doi: 10.1016/j.aml.2012.06.001.  Google Scholar

[5]

J. Bang-Jensen and G. Gutin, Digraphs. Theory, Algorithms and Applications, 2nd ed., Springer Verlag, London, 2009. doi: 10.1007/978-1-84800-998-1.  Google Scholar

[6]

N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall, Inc., Englewood Cliffs, 1974.  Google Scholar

[7]

B. Dorn, Flows in Infinite Networks - A Semigroup Aproach, Ph.D thesis, University of Tübingen, 2008. Google Scholar

[8]

B. Dorn, Semigroups for flows in infinite networks, Semigroup Forum, 76 (2008), 341-356. doi: 10.1007/s00233-007-9036-2.  Google Scholar

[9]

B. Dorn, M. Kramar Fijavž, R. Nagel and A. Radl, The semigroup approach to transport processes in networks, Physica D, 239 (2010), 1416-1421. doi: 10.1016/j.physd.2009.06.012.  Google Scholar

[10]

B. Dorn, V. Keicher and E. Sikolya, Asymptotic periodicity of recurrent flows in infinite networks, Math. Z., 263 (2009), 69-87. doi: 10.1007/s00209-008-0410-x.  Google Scholar

[11]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000.  Google Scholar

[12]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, Inc. New York, 1959.  Google Scholar

[13]

Ch. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0163-9.  Google Scholar

[14]

F.M. Hante, G. Leugering and T. I. Seidman, Modeling and analysis of modal switching in networked transport systems, Appl. Math. & Optimization, 59 (2009), 275-292. doi: 10.1007/s00245-008-9057-6.  Google Scholar

[15]

M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3.  Google Scholar

[16]

T. Matrai and E. Sikolya, Asymptotic behaviour of flows in networks, Forum Math., 19 (2007), 429-461. doi: 10.1515/FORUM.2007.018.  Google Scholar

[17]

C. D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719512.  Google Scholar

[18]

H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.  Google Scholar

[19]

R. Nagel, ed., One-parameter Semigroups of Positive Operators, Springer Verlag, Berlin, 1986.  Google Scholar

[20]

P. Namayanja, Transport on Network Structures, Ph.D thesis, UKZN, 2012. Google Scholar

[21]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[22]

E. Sikolya, Semigroups for Flows in Networks, Ph.D dissertation, University of Tübingen, 2004. Google Scholar

[23]

E. Sikolya, Flows in networks with dynamic ramification nodes, J. Evol. Equ., 5 (2005), 441-463. doi: 10.1007/s00028-005-0221-z.  Google Scholar

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