American Institute of Mathematical Sciences

Advanced
April  1998, 4(2): 273-300. doi: 10.3934/dcds.1998.4.273

Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation

Denis Mercier 1, and  Serge Nicaise 2,

1. 

Université de Valenciennes et du Hainaut Cambrésis, Limav, B.P. 311, 59304 Valenciennes Cedex, France
2. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9, France

Received  April 1997 Revised  June 1997 Published  February 1998

In this paper, we first prove existence results for general systems of differential equations of parabolic and hyperbolic type in a Hilbert space setting using the notion of Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary and sufficient condition on the boundary and/or transmission conditions which insures the dissipativity of the (spatial) operators. Our second goal is to take advantage of the one-dimensional structure of networks in order to build appropriate prewavelet bases in view to the numerical approximation of the above problems. Indeed we show that the use of such bases for their approximation (by the Galerkin method for elliptic operators and a fully discrete scheme for parabolic ones) leads to linear systems which can be preconditioned by a diagonal matrix and then can be reduced to systems with a condition number uniformly bounded (with respect to the mesh parameter).
Keywords: systems, Networks, differential equations, prewavelets..
Mathematics Subject Classification: 35G10, 35E20, 65N5.
Citation: Denis Mercier, Serge Nicaise. Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 273-300. doi: 10.3934/dcds.1998.4.273
[1]

Marissa Condon, Alfredo Deaño, Arieh Iserles. On systems of differential equations with extrinsic oscillation. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1345-1367. doi: 10.3934/dcds.2010.28.1345
[2]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609
[3]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
[4]

Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765
[5]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[6]

Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078
[7]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861
[8]

Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51
[9]

Józef Banaś, Monika Krajewska. On solutions of semilinear upper diagonal infinite systems of differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 189-202. doi: 10.3934/dcdss.2019013
[10]

Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
[11]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281
[12]

Maoxing Liu, Yuming Chen. An SIRS model with differential susceptibility and infectivity on uncorrelated networks. Mathematical Biosciences & Engineering, 2015, 12 (3) : 415-429. doi: 10.3934/mbe.2015.12.415
[13]

Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
[14]

Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193
[15]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[16]

Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
[17]

A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373
[18]

Elena Goncharova, Maxim Staritsyn. On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1061-1070. doi: 10.3934/dcdss.2018061
[19]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[20]

Uwe Helmke, Michael Schönlein. Minimum sensitivity realizations of networks of linear systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 241-262. doi: 10.3934/naco.2016010

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences