# American Institute of Mathematical Sciences

July  2018, 23(5): 1873-1893. doi: 10.3934/dcdsb.2018185

## Generalized network transport and Euler-Hille formula

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa 2 Institute of Mathematics, Łódź University of Technology, Łódź, Poland 3 Advanced System Analysis Program, International Institute for Applied System Analysis, Laxenburg, Austria

* Corresponding author: Jacek Banasiak

The research was partially conducted during the scholarship of A. P. at the International Institute for Applied System Analysis and supported by a grant for young scientists of the Institute of Mathematics of Lódź University of Technology. J. B. was partially supported by the Incentive Funding of the National Research Foundation of South Africa.

Received  April 2017 Revised  August 2017 Published  May 2018

In this article we consider asymptotic properties of network flow models with fast transport along the edges and explore their connection with an operator version of the Euler formula for the exponential function. This connection, combined with the theory of the regular convergence of semigroups, allows for proving that for fast transport along the edges and slow rate of redistribution of the flow at the nodes, the network flow semigroup (or its suitable projection) can be approximated by a finite dimensional dynamical system related to the boundary conditions at the nodes of the network. The novelty of our results lies in considering more general boundary operators than that allowed for in previous papers.

Citation: Jacek Banasiak, Aleksandra Puchalska. Generalized network transport and Euler-Hille formula. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1873-1893. doi: 10.3934/dcdsb.2018185
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##### References:
Commutativity of the aggregation diagram
The graph G representing the canal network in Example 1
The line graph of the graph shown on Fig. 2
Graphical representation of the Kimmel–Stivers model
Kimmel–Stievers model with vital dynamics
Discrete Lebowitz–Rubinow–Rotenberg model
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