Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks

• Published in: Journal of mathematical analysis and applications Vol. 362; no. 2; pp. 374 - 386
• Main Authors: D'Apice, C., Manzo, R., Piccoli, B.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.02.2010; Elsevier
• Subjects: BV estimates; Conservation laws; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Supply chains; Conservation laws; BV estimates; Supply chains; Existence of solution; State space; State space method; Partial differential equation; Wavefront; Cauchy problem; Conservation law; Mathematical analysis; Mixed problem; Approximate solution; Application
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2009.07.058

We consider a continuum-discrete model for supply chains based on partial differential equations. The state space is formed by a graph: The load dynamics obeys to a continuous evolution on each arc, while at nodes the good density is conserved, while the processing rate is adjusted. To uniquely determine the dynamics at nodes, the through flux is maximized, with the minimal possible processing rate change. Existence of solutions to Cauchy problems is proven. The latter is achieved deriving estimates on the total variation of the density flux, density and processing rate along a wave-front tracking approximate solution. Then the extension to supply networks is showed.