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Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics. (CAPA)
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.
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: 978-91-506-2269-0 (print)OAI: oai:DiVA.org:uu-161314DiVA: diva2:483681
Public defence
2012-03-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2005-3152
Available from: 2012-02-16 Created: 2011-11-10 Last updated: 2012-02-16Bibliographically approved
List of papers
1. A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition
Open this publication in new window or tab >>A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition
2011 (English)In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 16, 1227-1243 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-128863 (URN)10.1016/j.cnsns.2010.07.008 (DOI)000283826100014 ()
Funder
Swedish Research Council, 2005-3152
Available from: 2010-07-15 Created: 2010-07-28 Last updated: 2017-12-12Bibliographically approved
2. A computer-assisted proof of the existence of traveling wave solutions to the scalar Euler equations with artificial viscosity
Open this publication in new window or tab >>A computer-assisted proof of the existence of traveling wave solutions to the scalar Euler equations with artificial viscosity
2012 (English)In: NoDEA. Nonlinear differential equations and applications (Printed ed.), ISSN 1021-9722, E-ISSN 1420-9004, Vol. 19, 97-131 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-154660 (URN)10.1007/s00030-011-0120-7 (DOI)000299520600006 ()
Funder
Swedish Research Council, 2005-3152
Available from: 2011-06-02 Created: 2011-06-08 Last updated: 2017-12-11Bibliographically approved
3. Estimation of the diffusion coefficient by interval methods, level curves and bicubic splines
Open this publication in new window or tab >>Estimation of the diffusion coefficient by interval methods, level curves and bicubic splines
(English)Article in journal (Other academic) Submitted
Abstract [en]

We propose a new method for estimating a solution dependent diffusion coefficient in the heat equation, given a numerical solution to the latter. The main idea is to use a set–valued approximation of the solution in order to construct constraints on the coefficient. These constraints enable us to obtain a cover of the graph of the diffusion coefficient for a discrete set of temperatures. We illustrate the pros and cons of our method on several examples.

Keyword
inverse problem, bicubic spline, interval analysis, heat equation
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-161311 (URN)
Funder
Swedish Research Council, 2005-3152
Available from: 2012-01-25 Created: 2011-11-10 Last updated: 2012-03-16Bibliographically approved
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