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Mathematical modelling on interaction between malaria parasites and the host immune systemMarijani, Theresia 12 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / Please refer to full text for abstract.
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Comparative analysis of predictive equations for transfer processes in different porous structuresWoudberg, Sonia 12 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Research on transfer processes in various types of porous media has become important
for the optimization of high technology engineering devices and processes. In this study
the micro-structural parameters of different types of porous media, namely granular media,
foamlike media and fibre beds, are characterized and quantified. Existing analytical
modelling procedures for the three different types of porous media have been unified and
refined to improve their predictive capabilities. Deterministic equations are proposed for
predicting the streamwise pressure gradient, permeability and inertial coefficient of each
type of porous medium. The equations are applicable over the entire porosity range and
steady laminar flow regime and well suited as drag models in numerical computations.
It is shown that the improved granular model can be regarded as qualitative and quantitative
proof of the extensively used semi-empirical Ergun equation. The proposed model
is used to provide physical meaning to the empirical coefficients. An Ergun-type equation
is also proposed for foamlike media by remodelling the interstitial geometric configuration
and accompanying flow conditions.
The range of applicability of the existing foam model has been extended by incorporating
the effect of developing flow in the pressure drop prediction. An equation is proposed
in which the variation in the cross-sectional shape of the fibres can be incorporated into
the interstitial form drag coefficient used in the foam model. This serves as an improvement
on the constant value previously used. The existing foam model is also adapted
to account for anisotropy resulting from compression. Two case studies are considered,
namely compression of a non-woven glass fibre filter and compression of a soft polyester
fibre material. The significant effect of compression on permeability is illustrated. In
each case study the permeability values range over more than an order of magnitude for
the narrow porosity ranges involved. The pressure drop prediction of the foam model is
furthermore adapted to account for the combined effects of compression and developing
flow. The newly proposed model diminishes the significant over-prediction of the existing
foam model.
An equation is furthermore proposed for predicting the permeability of Fontainebleau
sandstones in which the effect of blocked throats is accounted for. Lastly, equations are
proposed for predicting diffusivity ratios of unconsolidated arrays of squares and cubes.
The prediction of the diffusivity ratio proposed in the present study, as opposed to model
predictions from the literature, takes into account diffusion that may take place in stagnant
fluid volumes. It is shown that a specific weighted average model proposed in the literature
is not adequate to predict the diffusivity ratio of fully staggered arrays of squares, since it is
shown not to be applicable to rectangular unit cells. Instead a new weighted average model
is proposed which is applicable over the entire porosity range and for both staggered and
non-staggered arrays of solid squares and cubes. The proposed weighted average model
provides satisfactory agreement with experimental data from the literature and numerical
data generated in the present study. / AFRIKAANSE OPSOMMING: Navorsing op oordragsprosesse in verskeie tipes poreuse media het belangrik geword vir die
optimisering van ho¨e-tegnologie ingenieurstoestelle- en prosesse. In hierdie studie word die
mikro-struktuur parameters van verskillende tipes poreuse media, naamklik korrelagtige
media, sponsatige media en veselbeddens gekarakteriseer en gekwantifiseer. Bestaande
analitiese modelleringsprosedures vir die drie verskillende tipes poreuse media is verenig
en verfyn om die voorspelbare bekwaamheid daarvan te verbeter. Deterministiese vergelykings
is voorgestel vir die voorspelling van die stroomsgewyse gradi¨ent, permeabiliteit en
inersi¨ele ko¨effisi¨ent van elke tipe poreuse medium. Die vergelykings is geldig oor die hele
porositeitsgrens en gestadigde laminˆere vloeigrens en goed geskik as weerstandsmodelle
in numeriese berekeninge.
Dit is aangetoon dat die verbeterde korrelmodel beskou kan word as kwalitatiewe en
kwantitatiewe bewys van die ekstensiewe gebruikte semi-empiriese Ergun vergelyking. Die
voorgestelde model is gebruik om fisiese betekenis aan die empiriese ko¨effisi¨ente te gee. ’n
Ergun-tipe vergelyking is ook voorgestel vir sponsagtige media deur hermodellering van
die tussenruimtelike geometriese konfigurasie en gepaardgaande vloeikondisies.
Die grense van toepaslikheid van die bestaande sponsmodel is uitgebrei deur die inkorporering
van die effek van ontwikkelende vloei in die voorspelling van die drukval. ’n
Vergelyking is voorgestel waarin die variasie in die deursnit vorm van die vesels ingesluit is
in die sponsmodel. Dit dien as verbetering op die konstante waarde wat voorheen gebruik
is. Die bestaande sponsmodel is ook aangepas om voorsiening te maak vir anisotropie
as gevolg van kompressie. Twee gevallestudies is oorweeg, naamlik kompressie van ’n
nie-geweefde glasvesel filter en kompressie van ’n sagte polyester veselmateriaal. Die
beduidende effek van kompressie op permeabiliteit is aangetoon. In elke gevallestudie
strek die permeabiliteit waardes oor meer as ’n grootte orde vir die skrale porositeitgrense
betrokke. Die drukvalvoorspelling van die sponsmodel is verder aangepas om voorsiening
te maak vir die gekombineerde effekte van kompressie en ontwikkelende vloei. Die
nuwe voorgestelde model verminder die beduidende oor-voorspelling van die bestaande
sponsmodel.
’n Vergelyking is verder voorgestel vir die voorspelling van die permeabiliteit van Fontainebleau
sandsteen waarin die effek van geblokte porie¨e in ag geneem is. Laastens is vergelykings
voorgestel vir die voorspelling van die diffusiwiteitsverhoudings van nie-konsoliderende
rangskikkings van vierkante en kubusse. Die diffusiwiteitsverhouding voorspel in die
huidige studie, teenoor modelvoorspellings vanaf die literatuur, neem diffusie in ag wat
plaasvind in die stagnante vloeistofvolumes. Dit is aangetoon dat ’n geweegde gemiddelde
model, voorgestel in die literatuur, nie in staat is om die diffusiwiteitsverhouding
van ten volle verspringende rangskikkings van vierkante te voorspel nie, aangesien dit nie
toepaslik is vir reghoekige eenheidselle nie. ’n Nuwe geweegde model is in plaas daarvan
voorgestel wat toepaslik is oor die hele porositeitsgrens en vir beide verspringende en nieverspringende
rangskikkings van soliede vierkante en kubusse. Die voorgestelde geweegde
gemiddelde model bied bevredigende ooreenstemming met eksperimentele data uit die
literatuur en numeriese data gegenereer in die huidige studie.
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An analogue of the Andre-Oort conjecture for products of Drinfeld modular surfacesKarumbidza, Archie 03 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura
varieties. In our case we consider the André-Oort conjecture for special points in the
product of Drinfeld modular varieties. We in particular manage to prove the André-
Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under
a characteristic assumption. / AFRIKAANSE OPSOMMING: Hierdie tesis handel van 'n funksieliggaam analoog van die André-Oort Vermoeding.
Die (Klassieke) André-Oort Vermoeding het betrekking tot die verspreiding van
spesiale punte op Shimura varietiete. Ons geval beskou ons die André-Oort Vermoeding
vir spesiale punte op die produk Drinfeldse modulvarietiete. In die besonders,
bewys ons die André-Oort Vermoeding vir ondervarieteite van 'n produk van twee
Drinfeldse modulvarietiete, onderhewig aan 'n karakteristiek-aanname.
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Rank matrix cascade algorithm, hermite interpolationDongmo, Guy Blaise 12 1900 (has links)
Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, into
powerful mathematical tools, for example in signal analysis and geometric modelling.
Both wavelet and subdivision analysis are based on the concept of a matrix–refinable
function, i.e. a finitely supported matrix function which is self-replicating in the sense
that it can be expressed as a linear combination of the integer shifts of its own dilation
with factor 2:
F = TAF = å
k∈Z
F(2 ・ −k)Ak.
The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute the
so-called matrix- mask sequence. Wavelets are in fact constructed as a specific linear
combination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2;
9], whereas the convergence of the associated matrix- subdivision scheme
c0 = c, cr+1 = SAcr, r ∈ Z+,
SA : c = (ck : k ∈ Z) 7→ SAc =
å
ℓ∈Z
Ak−2ℓ cℓ : k ∈ Z
!
,
subject to the necessary condition that
rank := dim
\
ǫ∈{0,1}
n
y ∈ Rd : Qǫy = y
o
> 0, Qǫ := å
j∈Z
Aǫ+2j, ǫ ∈ {0, 1},
( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinable
with respect to the mask coefficients defining the refinement equation and the
subdivision scheme.
Throughout this thesis, we investigate in time–domain for a given matrix mask sequence,
the related issues of the existence of a matrix–refinable function and the convergence
of the corresponding matrix– cascade algorithm, and finally we apply some
results to the particular research area of Hermite interpolatory subdivision schemes.
The dissertation is organized as follows:
In order to provide a certain flexibility or freedom over the project, we established
in Chapter 1 the equivalence relation between the matrix cascade algorithm and the
matrix subdivision scheme, subject to a well defined class of initial iterates.
Despite the general noncommutativity of matrices, we make use in the full rank case
Qǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some useful
tools, yielding a convergence result which comes as close to the scalar case as possible:
we obtained a concrete sufficient condition on the mask sequence based on the matrix
version of the generating function introduced in [3, page 22] for existence and convergence.
Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29],
our result on scalar case provided a progress for general mask sequences. We then
applied to obtain a new one-parameter family of refinable functions which includes
the cardinal splines as a special case, as well as corresponding convergent subdivision
schemes.
With the view to broaden the class of convergent matrix-masks, we replaced in chapter
3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u :=
(1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theory
of rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvement
with a generalization of [13, Theorem 13, p.8] in to the context of rank subdivision
schemes.
In Chapter 4, we translated the concrete convergence criteria of the general theory from
Theorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], into
the context of rank, factorization and spectral radius (cf. [26]), and presented a careful
analysis of the relationship between the two concepts. We then proceed with generalizations
and improvements: we classified the matrix cascade algorithms in term of
rank = 1, 2, . . . , d, and provided a complete characterization of each class with the use
of a more general r−norming factor namely τ(r)-norming factor. On the other hand,
we presented numerical methods to determine, if possible, the convergence of each
class of matrix cascade algorithms.
In both the scalar and matrix cases above, we also obtained explicitly the geometric
constant appearing in the estimate for the geometric convergence of thematrix-cascade
algorithm iterates to the matrix- refinable function. This same geometric convergence
rate therefore also holds true for the corresponding matrix–cascade algorithm.
Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 to
the particular research area of Hermite interpolatory subdivision schemes: we provided
a new convergence criterium, and end up with new convergence ranges of the
parameters’ values of the famous Hermite interpolatory subdivision scheme with two
parameters, due to Merrien [23]. / AFRIKAANSE OPSOMMING :(Wiskundige simbole het verander) Golfie en subdivisietegnieke het oor die afgelope twee dekades ontwikkel in kragtige wiskundige gereedskap, byvoorbeeld in seinanalise en geometriesemodellering. Beide
golfie en subdivisie analise is gebaseer op die konsep van ’n matriks-verfynbare funksie;
oftewel ’n eindig-ondersteunde matriksfunksie F wat selfreproduserend is in die sin
dat dit uitgedruk kan word as ’n lineêre kombinasie van die heelgetalskuiwe van F se
eie dilasie met faktor 2:
F = Σ F(2 · −α)A(α), met A(α), α ∈ Z, wat aandui die sogenaamde matriks-masker ry. Golfies kan dan gekonstrueer word as ’n spesifieke lineêre kombinasie van die
funksie ry {F(2 · −α) : α ∈ Z} (sien [2; 9]), terwyl die konvergensie van die ooreenstemmende
matriks-subdivisie skema
cº = c, cr+1 =(Σ
β∈Z A(α − 2β) cr(β) : α ∈ Z
!
, r ∈ Z+,
onderhewig aan die nodige voorwaarde dat rank := dim
\
ǫ∈{0,1}
n
y ∈ Rd : Qǫy = y
o
> 0, Qǫ := å
α∈Z
A(ǫ + 2α), ǫ ∈ {0, 1},
(sien [27]) die bestaan impliseer van ’n eindig-ondersteunde matriksfunksie F wat
verfynbaar ismet betrekking tot diemaskerko¨effisi¨entewat die subdivisieskema definieer,
en in terme waarvan die limietfunksie F van die subdivisieskema uitgedruk kan word
as F = å
α∈Z
F(· − α)c(α).
Ons hoofdoel hier is om , in die tydgebied, en vir ’n gegewematriks-masker ry, die verwante
kwessies van die bestaan van ’nmatriks-verfynbare funksie en die konvergensie
van die ooreenstemmende matriks-kaskade algoritme, en matriks-subdivisieskema, te
ondersoek, en om uiteindelik sommige van ons resultate toe te pas op die spesifieke
kwessie van die konvergensie van Hermite interpolerende subdivisieskemas.
Summary v
Eerstens, in Hoofstuk 1, ondersoek ons die verwantskap tussen matriks-kaskade algoritmes
en matriks-subdivisie skemas, met verwysing na ’n goedgedefinieerde klas van
begin-iterate.
Vervolgens beskou ons die volle rang geval Qǫ = I, ǫ ∈ {0, 1}, om, in Hoofstuk 2,
nuttige gereedskap te ontwikkel, en wat daarby ’n konvergensie resultaat met ’n sterk
konneksie ten opsigte van die skalaar-geval oplewer.
Met die doelstelling om ons klas van konvergente matriks-maskers te verbreed, vervang
ons, in Hoofstuk 3, die volle rang voorwaarde met die rang een voorwaarde Qǫu =
u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, en verkry ons dan ’n verbetering op ’n konvergensieresultaat
in die artikel [14] deur Dubuc en Merrien, deur gebruik te maak van die
teorie van rang subdivisieskemas van Micchelli en Sauer [26; 27], waarna ons die resultaat
[14, Stelling 13, page 8] na die konteks van rang subdivisieskemas veralgemeen.
InHoofstuk 4 herlei ons die konkrete konvergensie kriteria van Stelling 3.2, soos gebaseer
op die r-normerende faktor gedefinieer in [14, Definisie 6, page 6] , na die konteks
van rang, faktorisering en spektraalradius (sien [27]), en gee ons ’n streng analise
van die verwantskap tussen die twee konsepte. Verder stel ons dan bekend ’n nuwe
klassifikasie van matriks-kaskade algoritmes ten opsigte van rang, en verskaf ons ’n
volledige karakterisering van elke klasmet behulp van ’nmeer algemene r-normerende
faktor, nl. die τ(r)-normerende faktor. Daarby gee ons doeltreffende numeriesemetodes
vir die implementering van ons teoretiese resultate.
Ons verkry ook eksplisiet die geometriese konstante wat voorkom in die afskatting
van die geometriese konvergensie van die matriks-kaskade algoritme iterate na die
matriks-verfynbare funksie.
Ten slotte, in Hoofstuk 5, pas ons die teorie en algoritmes ontwikkel in Hoofstuk 4 toe
om die konvergensie van Hermite-interpolerende subdivisieskemas te analiseer. Spesifiek
lei ons ’n nuwe konvergensie kriterium af, wat ons dan toepas om nuwe konvergensie
gebiede vir die parameter waardes te verkry vir die beroemde Hermite interpolerende
subdivisieskema met twee parameters, soos toegeskryf aan Merrien [24].
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On the coefficients of Drinfeld modular forms of higher rankBasson, Dirk Johannes 04 1900 (has links)
Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and
while it is known that a higher rank theory could be possible, higher rank
Drinfeld modular forms have only recently been de ned. In 1988 Gekeler
published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular
forms. The goal of this thesis is to perform a similar study of the coe cients
of higher rank Drinfeld modular forms.
The main results are that the coe cients themselves are (weak) Drinfeld
modular forms, a product formula for the discriminant function, the rationality
of certain naturally de ned modular forms, and the computation of
some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer
en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër
rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler
die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van
rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is
om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer.
Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre
vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere
natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van
Hecke eievorme en hul eiewaardes.
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Interest rates market and models after the 2007 credit crunchRahantamialisoa, Tahirivonizaka Fanirisoa Zazaravaka 03 1900 (has links)
Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The interest rates market has changed dramatically since the 2007 credit
crunch with the explosion of basis spreads between rates of different tenors
and currencies. Consequently, the classical replication of FRA rates with spot
LIBOR rates is no longer valid. Moreover, the 2007 credit crunch yields a separation
between the curve used for discounting and the forward or projection
curves that estimate all future cash-fl ows. Another impact of the credit crunch
in risk management is that market participants have started to give more importance
to the difference between collateralized and uncollateralized trades.
Nowadays, the wide spread use of collateral, especially in swap contracts, has
made the overnight index swap (OIS) rate the appropriate benchmark for
discounting collateralized trades. Inspired by the seminal works of Mercurio
(2010a,b), Kijima et al. (2008), Fujii et al. (2011), Bianchetti (2010b), with
the contributions of other authors, and motivated by the evolution of the interest
rates market and models, this thesis examines a new framework that uses
multiple-curves to value interest rate derivatives which is compatible with the
current market practice.
Firstly, we discuss the roots of the 2007 credit crunch and its serious consequences
for pricing interest rate derivatives. We underscore the necessity of a
multiple-curve pricing framework for interest rate derivatives. This is followed
by a discussion on the importance of collateralization and OIS discounting in
pricing Over-The-Counter (OTC) derivatives. The central part of the thesis
discusses the modern theoretical framework and the practical implementation of the multiple curve pricing method. We present a bootstrapping algorithm
used to construct and fit the multiple-yield curves to market prices of plainvanilla
contracts.
Secondly, starting with the single-currency economy, the extended version
of the LIBOR Market Model, developed by Mercurio (2010a,b), which proposes
a joint model of FRA rates, implied forward rates and their corresponding
spread is investigated. Analogously, the extended version of short-rate model
in a multiple-curve setup and in the presence of basis spread, proposed by
Kijima et al. (2008), is presented and discussed. This work provides a detailed
analysis of these extensions and the corresponding closed formulae for liquid
products such as caps and swaptions. Finally, in the multiple-currencies case,
the HJM model with stochastic basis spreads, introduced by Fujii et al. (2011),
consistent with the foreign exchange and cross-currency swaps markets that
includes the effect of collateralization is examined thoroughly. / AFRIKAANSE OPSOMMING: Die rentekoers mark het dramaties verander sedert die 2007 krediet krisis met
'n ontplo ng van basisverspreidings tussen koerse van verskillende looptye
("tenor") en geldeenhede. As gevolg, is die klassieke replikasie van FRA koerse
met LIBOR sigkoerse nie langer geldig nie. Verder het die 2007 kredietkrisis
'n skeiding veroorsaak tussen die kromme wat gebruik word vir diskontering
en die voorwaardse of vooruitskattings krommes wat toekomstige kontantvloei
voorspel. 'n Verdere impak van die kredietkrisis in risikobestuur is dat mark
deelnemers begin het om meer klem te lê op verskille tussen aangevulde en
onaangevulde handel. Deesdae, met die algemene gebruik van kollaterale sekuriteit, veral in ruiltransaksiekontrakte,
is die oornagse indeks ruiltransaksie (overnight index swap, OIS) koers die geskikte maatstaf om aangevulde handel te diskonteer. Geïnspireer
deur die gedagteryke werk van Mercurio (2010a,b), Kijima et al. (2008),
Fujii et al. (2011), Bianchetti (2010b), met bydrae van menige outeurs, en
gemotiveer deur die evolusie van die rentekoers markte en modelle, ondersoek
hierdie tesis 'n nuwe raamwerk wat multikrommes gebruik om rentekoers
afgeleide effekte te waardeer wat versoenbaar is met die lopende mark praktyk.
Eerstens, bespreek ons die oorsake van die 2007 kredietkrisis en die ernstige
nagevolge vir die waardering van rentekoers afgeleide effekte. Ons beklemtoon
die noodsaaklikheid van 'n multikromme waarderings raamwerk vir rentekoers
afgeleide effekte. Dit word gevolg deur 'n bespreking oor die belangrikheid
van aanvulling en OIS diskontering in die waardering van oor-die-toonbank (over-the-counter, OTC) effekte. Die teoretiese raamwerk en die praktiese
implimentering van die multikromme waarderings metode word bespreek. Ons
stel ook ten toon 'n skoenlus ("bootstrapping") algoritme wat gebruik kan word
om meervoudige opbrengs krommes saam te stel en die dan te pas op mark
pryse van vanielje kontrakte.
Tweedens, met 'n enkel geldeenheid ekonomie as beginpunt, word die uitgebreide
weergawe van die LIBOR Mark Model (ontwikkel deur Mercurio
(2010a,b), wat 'n gesamentlike model van FRA koerse voorstel), geïmpliseerde
termyn koerse en hul ooreenstemmende verspreiding bestudeer. Ooreenkomstig
word die uitgebreide weergawe van die kort koers model in 'n multikromme
opset en in die aanwesigheid van basisspreiding (voorgestel deur Kijima
et al. (2008)) uiteengesit en bespreek. Hierdie werk verskaf 'n uitvoerige
analise van hierdie uitbreidings en die ooreenstemmende geslote formules
vir vloeibare produkte soos perke en ruiltransaksie opsies. Ten slotte, in die
multi-geldeenheid geval, word die HJM model met stogastiese basisverspreiding
(voorgestel deur Fujii et al. (2011)), nie-strydig met buitelandse valuta en
kruisvaluta ruiltransaksie markte wat die effekte van aanvulling insluit word
deuglik bestudeer.
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Lie group analysis of exotic options.Okelola, Michael. 19 June 2014 (has links)
Exotic options are derivatives which have features that makes them more complex
than vanilla traded products. Thus, finding their fair value is not always an easy
task. We look at a particular example of the exotic options - the power option - whose
payoffs are nonlinear functions of the underlying asset price. Previous analyses of
the power option have only obtained solutions using probability methods for the
case of the constant stock volatility and interest rate. Using Lie symmetry analysis
we obtain an optimal system of the Lie point symmetries of the power option PDE
and demonstrate an algorithmic method for finding solutions to the equation. In
addition, we find a new analytical solution to the asymmetric type of the power
option.
We also focus on the more practical and realistic case of time dependent parameters:
volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new
exact solution for the terminal pay off case.
We also consider the power parameter of the option as a time dependent factor. A
new solution is once again obtained for this scenario. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.
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First integrals for the Bianchi universes : supplementation of the Noetherian integrals with first integrals obtained by using Lie symmetries.Pantazi, Hara. January 1997 (has links)
No abstract available. / Thesis (M.Sc.)-University of Natal, 1997.
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Ermakov systems : a group theoretic approach.Govinder, Kesh S. January 1993 (has links)
The physical world is, for the most part, modelled using second order ordinary differential equations. The time-dependent simple harmonic oscillator and the Ermakov-Pinney equation (which together form an Ermakov system) are two examples that jointly and separately describe many physical situations. We study Ermakov systems from the point of view of the algebraic properties
of differential equations. The idea of generalised Ermakov systems is introduced and their relationship to the Lie algebra sl(2, R) is explained. We show that the 'compact' form of generalized Ermakov systems has an infinite dimensional Lie algebra. Such algebras are usually associated only with first order equations in the context of ordinary differential equations. Apart from the Ermakov invariant which shares the infinite-dimensional algebra of the 'compact' equation, the other three integrals force the dimension of the algebra to be reduced to the three of sl(2, R). Subsequently we establish a new class of Ermakov systems by considering equations invariant under sl(2, R) (in two dimensions) and sl(2, R) EB so(3) (in three dimensions). The former class contains the generalized Ermakov system as a special case in which the force is velocity-independent. The latter case is a generalization of the classical equation of motion of the magnetic monopole
which is well known to possess the conserved Poincare vector. We demonstrate that in fact there are three such vectors for all equations of this type. / Thesis (M.Sc.)-University of Natal, 1993.
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Continuous symmetries of difference equations.Nteumagne, Bienvenue Feugang. 04 June 2013 (has links)
We consider the study of symmetry analysis of difference equations. The original work done by
Lie about a century ago is known to be one of the best methods of solving differential equations.
Lie's theory of difference equations on the contrary, was only first explored about twenty years
ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some
work has been done in the field of symmetries of difference equations for the past years. Given
an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to
analyze the problem. It is commonly known that the number of independent variables can be
reduced after the symmetries of the equation are obtained. One can determine the optimal
system of the equation in order to get a reduction of the independent variables. In addition,
using the method, one can obtain new solutions from known ones. This feature is interesting
because some differential equations have apparently useless trivial solutions, but applying Lie
symmetries to them, more interesting solutions are obtained.
The question arises when it happens that our equation contains a discrete quantity. In other
words, we aim at investigating steps to be performed when we have a difference equation. Doing
so, we find symmetries of difference equations and use them to linearize and reduce the order
of difference equations. In this work, we analyze the work done by some researchers in the field
and apply their results to some examples.
This work will focus on the topical review of symmetries of difference equations and going
through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.
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