Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
This PhD thesis treats some problems concerning nonlinear differential equations.
In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics.
Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines.
In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.
Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2012. , 21 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 76
computer-assisted proof, numerical verification, viscous Burgers’ equation, enclosure, existence, nonlinear boundary value problems, Euler equations, inverse problem, bicubic spline, interval analysis, heat equation, fluid limit, peer-to-peer networks, fixed points, stability, global stability
Research subject Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-161314ISBN: 978-91-506-2269-0OAI: oai:DiVA.org:uu-161314DiVA: diva2:483681
2012-03-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Zgliczynski, Piotr, Professor
Tucker, Warwick, ProfessorKreiss, Gunilla, Professor
FunderSwedish Research Council, 2005-3152
List of papers