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Some Problems in Kinetic Theory and Applications
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
2011 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This thesis consists of four papers. the first is devoted to discrete velocity models, the second to hydrodynamic equation beyond Navier-Stokes level, the third to a multi-linear Maxwell model for economic or social dynamics and the fourth is devoted to a function related to the Riemann zeta-function.

In Paper 1, we consider the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVM) of the classical Boltzman equation. We explain in detail how this problem can be solved and present a complete classification of normal plane DVMs with relatively small number n of velocities (n≤10). Some results for models with larger number of velocities are also presented.

In Paper 2, we discuss hydrodynamics at the Burnett level. Since the Burnett equations are ill-posed, we describe how to make a regularization of these. We derive the well-posed generalized Burnett equations (GBEs) and discuss briefly an optimal choice of free parameters and consider a specific version of these equations. Finally we prove linear stability for GBE and present some numerical result on the sound propagationbased on GBEs.

In Paper 3, we study a Maxwell kinetic model of socio-economic behavior. The model can predict a time dependent distribution of wealth among the participants in economic games with an arbitrary, but sufficiently large, number of players. The model depends on three different positive parameters {γ,q,s} where s and q are fixed by market conditions and γ is a control parameter. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented.

In Paper 4, we study a special function u(s,x), closely connected to the Riemann zeta-function ζ(s), where s is a complex number. We study in detail the properties of u(s,x) and in particular the location of its zeros s(x), for various x≥0. For x=0 the zeros s(0) coincide with non-trivial zeros of ζ(s). We perform a detailed numerical study of trajectories of various zeros s(x) of u(s,x).

Ort, förlag, år, upplaga, sidor
Karlstad: Karlstad University , 2011. , s. 22
Serie
Karlstad University Studies, ISSN 1403-8099 ; 2011:52
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
Matematik
Identifikatorer
URN: urn:nbn:se:kau:diva-8498ISBN: 978-91-7063-388-1 (tryckt)OAI: oai:DiVA.org:kau-8498DiVA, id: diva2:446956
Disputation
2011-12-01, 21A 342, Karlstads universitet, Karlstad, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2011-11-07 Skapad: 2011-10-10 Senast uppdaterad: 2011-11-07Bibliografiskt granskad
Delarbeten
1. Discrete velocity models of the Boltzmann equation and conservation laws
Öppna denna publikation i ny flik eller fönster >>Discrete velocity models of the Boltzmann equation and conservation laws
2010 (Engelska)Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, nr 1, s. 35-58Artikel i tidskrift (Refereegranskat) Published
Ort, förlag, år, upplaga, sidor
Springfield, MO: American Institute of Mathematical Sciences, 2010
Nationell ämneskategori
Matematik
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kau:diva-8709 (URN)10.3934/krm.2010.3.35 (DOI)000273989700004 ()
Tillgänglig från: 2011-11-03 Skapad: 2011-11-03 Senast uppdaterad: 2020-07-08Bibliografiskt granskad
2. Boltzmann equation and hydrodynamics at the Burnett level
Öppna denna publikation i ny flik eller fönster >>Boltzmann equation and hydrodynamics at the Burnett level
2012 (Engelska)Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, nr 2, s. 237-260Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.

Ort, förlag, år, upplaga, sidor
American Institute of Mathematical Sciences, 2012
Nyckelord
Hydrodynamics, regularized Burnett equations, Stability, sound propagation.
Nationell ämneskategori
Matematik
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kau:diva-8710 (URN)10.3934/krm.2012.5.237 (DOI)000302962700002 ()
Tillgänglig från: 2011-11-03 Skapad: 2011-11-03 Senast uppdaterad: 2020-01-03Bibliografiskt granskad
3. Kinetic modeling of economic games with large number of participants
Öppna denna publikation i ny flik eller fönster >>Kinetic modeling of economic games with large number of participants
2011 (Engelska)Ingår i: Kinetic and Related Models, ISSN 1937-5093, Vol. 4, nr 1, s. 169-185Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

                 We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters where is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.

Ort, förlag, år, upplaga, sidor
American Institute of Mathematical Sciences, 2011
Nyckelord
Maxwell models, self-similar solutions, distribution of wealth, market economy
Nationell ämneskategori
Matematik
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kau:diva-8711 (URN)10.3934/krm.2011.4.169 (DOI)000286926200010 ()
Tillgänglig från: 2011-11-03 Skapad: 2011-11-03 Senast uppdaterad: 2012-12-04Bibliografiskt granskad
4. On a special function related to the Riemann zeta-function
Öppna denna publikation i ny flik eller fönster >>On a special function related to the Riemann zeta-function
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
Nationell ämneskategori
Matematik
Forskningsämne
Matematik
Identifikatorer
urn:nbn:se:kau:diva-8712 (URN)
Tillgänglig från: 2011-11-03 Skapad: 2011-11-03 Senast uppdaterad: 2011-11-07Bibliografiskt granskad

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