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Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential EquationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics , 2012. , 21 p.
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 76
##### Keyword [en]

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
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-161314ISBN: 978-91-506-2269-0OAI: oai:DiVA.org:uu-161314DiVA: diva2:483681
##### Public defence

2012-03-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt384",{id:"formSmash:j_idt384",widgetVar:"widget_formSmash_j_idt384",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt390",{id:"formSmash:j_idt390",widgetVar:"widget_formSmash_j_idt390",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt396",{id:"formSmash:j_idt396",widgetVar:"widget_formSmash_j_idt396",multiple:true});
##### Funder

Swedish Research Council, 2005-3152
Available from: 2012-02-16 Created: 2011-11-10 Last updated: 2012-02-16Bibliographically approved
##### List of papers

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.

1. A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition$(function(){PrimeFaces.cw("OverlayPanel","overlay331893",{id:"formSmash:j_idt432:0:j_idt436",widgetVar:"overlay331893",target:"formSmash:j_idt432:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. A computer-assisted proof of the existence of traveling wave solutions to the scalar Euler equations with artificial viscosity$(function(){PrimeFaces.cw("OverlayPanel","overlay421499",{id:"formSmash:j_idt432:1:j_idt436",widgetVar:"overlay421499",target:"formSmash:j_idt432:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Estimation of the diffusion coefficient by interval methods, level curves and bicubic splines$(function(){PrimeFaces.cw("OverlayPanel","overlay483515",{id:"formSmash:j_idt432:2:j_idt436",widgetVar:"overlay483515",target:"formSmash:j_idt432:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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