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Adam Błoch

Publications and source records attributed to Adam Błoch.

5 recordsLinked to original sources
arXiv2026

Network transport with nonlinear dynamics at the nodes

In this paper, we consider a network transport model in which agents moving along the edges can contribute to dynamics at nodes or bypass them. The model takes the form of a system of first-order partial differential equations coupled with a system of ordinary differential equations, and can describe a range of phenomena, from diseases in metapopulations to migratory systems with delays, to cell differentiation processes, providing a unified platform that includes network transport and delay systems as particular cases. We prove the well-posedness of the model in $ L^p$ spaces, $ 1\leq p<\infty$, study long-term asymptotics, and illustrate the theory using an SIS disease in a metapopulation consisting of several sites where the disease develops, connected by routes along which the population can migrate, as an example.

math.AP↗
arXiv2024

Existence and explicit formula for a semigroup related to some network problems with unbounded edges

In this paper we consider an initial-boundary value problem related to some network dynamics where the underlying graph has unbounded edges. We show that there exists a C0-semigroup for this problem using a general result from the literature. We also find an explicit formula for this semigroup. This is achieved using the method of characteristics and then showing that the Laplace transform of the solution is equal to the resolvent operator of the generator.

math.DS↗
arXiv2021

Telegraph type systems on networks and port-Hamiltonians. II. Graph realizability

Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of $2\times 2$ hyperbolic equations on a metric graph $\Gamma$. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $\Gamma$ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $\Gamma$.

math.DS↗
arXiv2021

Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posedness

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.

math.AP↗