The Nonlinear Heat Equation on W-Random Graphs

For systems of coupled differential equations on a sequence of W -random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limi...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2014-06, Vol.212 (3), p.781-803
Main Author: Medvedev, Georgi S.
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Continuums
Differential equations
Evolution
Fluid- and Aerodynamics
Graphs
Heat equations
Initial value problems
Mathematical and Computational Physics
Mathematical models
Nonlinearity
Physics
Physics and Astronomy
Theoretical
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Summary:For systems of coupled differential equations on a sequence of W -random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in Medvedev (The nonlinear heat equation on dense graphs and graph limits. ArXiv e-prints, 2013 ) justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-013-0706-9