Spelling suggestions: "subject:"themathematics."" "subject:"theirmathematics.""
151 |
Lie symmetries of junction conditions for radiating stars.Abebe, Gezahegn Zewdie. 31 October 2013 (has links)
We consider shear-free radiating spherical stars in general relativity. In particular we study the junction condition relating the pressure to the heat
flux at the boundary of the star. This is a nonlinear equation in the metric functions. We analyse the junction condition when the spacetime is conformally flat, and when the particles are travelling in geodesic motion. We transform the governing equation using the method of Lie analysis. The Lie symmetry generators that leave the equation invariant are identifed
and we generate the optimal system in each case. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. As a result, particular solutions to the junction condition are presented. These exact solutions can be presented in terms of elementary functions. Many of the solutions found are new and could be useful in the modelling process. Our analysis is the first comprehensive treatment of the boundary condition using a symmetry approach. We have shown that this approach is useful in generating new results. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.
|
152 |
Noether's theorem and first integrals of ordinary differential equations.Moyo, Sibusiso. January 1997 (has links)
The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach. / Thesis (M.Sc.)-University of Natal, Durban, 1997.
|
153 |
Fischer Clifford matrices and character tables of certain groups associated with simple groups O+10(2) [the simple orthogonal group of dimension 10 over GF (2)], HS and Ly.Seretlo, Thekiso Trevor. January 2011 (has links)
The character table of any finite group provides a considerable amount of information about a group and the use of character tables is of great importance in Mathematics and Physical Sciences. Most of the maximal subgroups of finite simple groups and their automorphisms are extensions of elementary abelian groups. Various techniques have been used to compute character tables, however Bernd Fischer came up with the most powerful and informative technique of calculating character tables of group extensions. This method is known as the Fischer-Clifford Theory and uses Fischer-Clifford matrices, as one of the tools, to compute character tables. This is derived from the Clifford theory. Here G is an extension of a group N by a finite group G, that is G = N.G. We then construct a non-singular matrix for each conjugacy class of G/N =G. These matrices, together with partial character tables of certain subgroups of G, known as the inertia groups, are used to compute the full character table of G. In this dissertation, we discuss Fischer-Clifford theory and apply it to both split and non-split extensions. We first, under the guidance of Dr Mpono, studied the group 27:S8 as a maximal subgroup of 27:SP(6,2), to familiarize ourselves to Fischer-Clifford theory. We then looked at 26:A8 and 28:O+8 (2) as maximal subgroups of 28:O+8 (2) and O+10(2) respectively and these were both split extensions. Split extensions have also been discussed quite extensively, for various groups, by
different researchers in the past. We then turned our attention to non-split extensions. We started with 24.S6 and 25.S6 which were maximal subgroups of HS and HS:2 respectively. Except for some negative signs in the first column of the Fischer-Clifford matrices we used the Fisher-Clifford theory as it is. The Fischer-Clifford theory, is also applied to 53.L(3, 5), which is a maximal subgroup of the Lyon's group Ly. To be able to use the Fisher-Clifford theory we had to consider projective representations and characters of inertia factor groups. This is not a simple method and quite some smart computations were needed but we were able to determine the character table of 53.L(3,5).
All character tables computed in this dissertation will be sent to GAP for incorporation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.
|
154 |
Spherically symmetric cosmological solutions.Govender, Jagathesan. January 1996 (has links)
This thesis examines the role of shear in inhomogeneous spherically symmetric spacetimes
in the field of general relativity. The Einstein field equations are derived for
a perfect fluid source in comoving coordinates. By assuming a barotropic equation
of state, two classes of nonaccelerating solutions are obtained for the Einstein field
equations. The first class has equation of state p = ⅓µ and the second class, with
equation of state p = µ, generalises the models of Van den Bergh and Wils (1985).
For a particular choice of a metric potential a new class of solutions is found which
is expressible in terms of elliptic functions of the first and third kind in general. A
class of nonexpanding cosmological models is briefly studied. The method of Lie
symmetries of differential equations generates a self-similar variable which reduces
the field and conservation equations to a system of ordinary differential equations.
The behaviour of the gravitational field in this case is governed by a Riccati equation
which is solved in general. Another class of solutions is obtained by making an ad
hoc choice for one of the gravitational potentials. It is demonstrated that for a stiff
fluid a particular case of the generalised Emden-Fowler equation arises. / Thesis (Ph.D.)-University of Natal, Durban, 1996.
|
155 |
Amplitude-shape method for the numerical solution of ordinary differential equations.Parumasur, Nabendra. January 1997 (has links)
In this work, we present an amplitude-shape method for solving evolution problems described
by partial differential equations. The method is capable of recognizing the special
structure of many evolution problems. In particular, the stiff system of ordinary differential
equations resulting from the semi-discretization of partial differential equations is considered.
The method involves transforming the system so that only a few equations are stiff
and the majority of the equations remain non-stiff. The system is treated with a mixed
explicit-implicit scheme with a built-in error control mechanism. This approach proved to
be very effective for the solution of stiff systems of equations describing spatially dependent
chemical kinetics. / Thesis (Ph.D.)-University of Natal, 1997.
|
156 |
Exact models for radiating relativistic stars.Rajah, Suryakumari Surversperi. January 2007 (has links)
In this thesis, we seek exact solutions for the interior of a radiating relativistic star undergoing gravitational collapse. The spherically symmetric interior spacetime, when matched with the exterior radiating Vaidya spacetime, at the boundary of the star, yields the governing equation describing the gravitational behaviour of the collapsing star. The investigation of the model hinges on the solution of the governing equation at the boundary. We first examine shear-free models which are conformally flat. The boundary condition is transformed to an Abel equation and several new solutions are generated. We then study collapse with shear in geodesic motion. Two classes of solutions are generated which are regular at the stellar centre. Our treatment extends the results of Naidu et al (2006) which had the undesirable feature of a singularity at the centre of the star. In an attempt to find more general models, we transform the fundamental equation to a Riccati equation. Two general classes of solution are found and are used to study the thermal evolution in the causal theory of thermodynamics. These solutions are shown to reduce to the Friedmann dust solution in the absence of heat flow. Furthermore, we obtain new categories of solutions for the case of gravitational collapse with expansion, shear and acceleration of the stellar fluid. This is achieved by transforming the boundary condition into a Riccati equation. In special cases the Bernoulli equation is regained. The solutions are given in terms of elementary functions and they permit the investigation of the physical features of radiative stellar collapse. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2007.
|
157 |
Polynomial approximations to functions of operators.Singh, Pravin. January 1994 (has links)
To solve the linear equation Ax = f, where f is an element of Hilbert space H and A
is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate
-1
the inverse operator A by an operator V which is a polynomial in A. Using the
spectral theory of bounded normal operators the problem is reduced to that of
approximating a function of the real variable by polynomials of best uniform
approximation. We apply two different techniques of evaluating
A-1 so that the
operator V is chosen either as a polynomial P (A) when P (A) approximates the
n n
function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn
(A)
approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A)
satisfy three point recurrence relations, thus the approximate solution vectors P (A)f
n
and Q (A)f can be evaluated iteratively. We compare the procedures involving
n
Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite.
We also show that the technique can be applied to an operator which is not selfadjoint,
but close, in the sense of operator norm, to a selfadjoint operator. The iterative
techniques we develop are used to solve linear systems arising from the discretization of
Freedholm integral equations of the second kind. Both smooth and weakly singular
kernels are considered. We show that earlier work done on the approximation of linear
functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse
operator and are thus special cases of a general result involving an approximation of
arbitrary degree to A -1 . / Thesis (Ph.D.)-University of Natal, 1994.
|
158 |
Conformally invariant relativistic solutions.Maharaj, M. S. January 1993 (has links)
The study of exact solutions to the Einstein and Einstein-Maxwell field equations,
by imposing a symmetry requirement on the manifold, has been the subject of much
recent research. In this thesis we consider specifically conformal symmetries in static
and nonstatic spherically symmetric spacetimes. We find conformally invariant solutions,
for spherically symmetric vectors, to the Einstein-Maxwell field equations
for static spacetimes. These solutions generalise results found previously and have
the advantage of being regular in the interior of the sphere. The general solution to
the conformal Killing vector equation for static spherically symmetric spacetimes is
found. This solution is subject to integrability conditions that place restrictions on
the metric functions. From the general solution we regain the special cases of Killing
vectors, homothetic vectors and spherically symmetric vectors with a static conformal
factor. Inheriting conformal vectors in static spacetimes are also identified. We
find a new class of accelerating, expanding and shearing cosmological solutions in
nonstatic spherically symmetric spacetimes. These solutions satisfy an equation of
state which is a generalisation of the stiff equation of state. We also show that this
solution admits a conformal Killing vector which is explicitly obtained. / Thesis (Ph.D.)-University of Natal, Durban, 1993.
|
159 |
Coherent structures and symmetry properties in nonlinear models used in theoretical physics.Harin, Alexander O. January 1994 (has links)
This thesis is devoted to two aspects of nonlinear PDEs which are fundamental
for the understanding of the order and coherence observed in the underlying physical
systems. These are symmetry properties and soliton solutions. We analyse these
fundamental aspects for a number of models arising in various branches of theoretical
physics and appli ed mathematics.
We start with a fluid model of a plasma in the case of a general polytropic
process. We propose a method of the analysis of unmagnetized travelling structures,
alternative to the conventional formalism of Sagdeev 's pseudopotential. This method
is then utilized to obtain the existence domain for compressive solitons and to establish
the absence of rarefactive solitons and monotonic double layers in a two-component
plasma.
The second class of models under consideration arises in (2+1)-dimensional condensed
matter physics. These are the Abelian gauge theories with Chern-Simons
term, which are currently considered as candidates for the description of high-Te
superconductivity and fra ctional quantum Hall effect. The emphasis here is on nonrelativistic
theories. The standard model of a self-gravitating gas of nonrelativistic
bosons coupled to the Chern-Simons gauge field is capable of describing asymptotically
vanishing field configurations , such as lump-like solitons. We formulate an
alternative model, which describes systems of repulsive particles with a background
electric charge and allows to incorporate asymptotically nonvanishing configurations,
such as condensate and its topological excitations. We demonstrate the absence of the condensate state in the standard nonrelativistic gauge theory and relate this fact
to the inadequate Lagrangian formulation of its nongauged precursor. Using an appropriate
modification of this Lagrangian as a basis for the gauge theory naturally
leads to the new model. Reformulating it as a constrained Hamiltonian system allows
us to find two self-duality limit s and construct a large variety of self-dual solutions.
We demonstrate the equivalence of the model with the background charge and the
standard model in the external magnetic field. Finally we discuss nontopological
bubble solutions in Chem-Simons-Maxwell theories and demonstrate their absence
in nonrelativistic theories.
Finally, we consider a model of a nonhomogeneous nonlinear string. We continue
the group theoretical classification of the string equations initiated by Ibragimov et
al. and present their preliminary group classification with respect to a countable dimensional subalgebra of their equivalence algebra. This subalgebra is an extension
of the 10-dimensional subalgebra considered by Ibragimov et al. Our main result here
is a table of non-equivalent equations possessing an additional symmetry. / Thesis (Ph.D.)-University of Natal, 1994.
|
160 |
Algebraic properties of ordinary differential equations.Leach, Peter Gavin Lawrence. January 1995 (has links)
In Chapter One the theoretical basis for infinitesimal transformations is
presented with particular emphasis on the central theme of this thesis which
is the invariance of ordinary differential equations, and their first integrals,
under infinitesimal transformations. The differential operators associated with
these infinitesimal transformations constitute an algebra under the operation
of taking the Lie Bracket. Some of the major results of Lie's work are recalled.
The way to use the generators of symmetries to reduce the order of a differential
equation and/or to find its first integrals is explained. The chapter concludes
with a summary of the state of the art in the mid-seventies just before the
work described here was initiated.
Chapter Two describes the growing awareness of the algebraic properties of
the paradigms of differential equations. This essentially ad hoc period demonstrated
that there was value in studying the Lie method of extended groups
for finding first integrals and so solutions of equations and systems of equations.
This value was emphasised by the application of the method to a class of
nonautonomous anharmonic equations which did not belong to the then pantheon
of paradigms. The generalised Emden-Fowler equation provided a route
to major development in the area of the theory of the conditions for the linearisation
of second order equations. This was in addition to its own interest.
The stage was now set to establish broad theoretical results and retreat from
the particularism of the seventies.
Chapters Three and Four deal with the linearisation theorems for second
order equations and the classification of intrinsically nonlinear equations according
to their algebras. The rather meagre results for systems of second
order equations are recorded.
In the fifth chapter the investigation is extended to higher order equations
for which there are some major departures away from the pattern established
at the second order level and reinforced by the central role played by these
equations in a world still dominated by Newton. The classification of third
order equations by their algebras is presented, but it must be admitted that
the story of higher order equations is still very much incomplete.
In the sixth chapter the relationships between first integrals and their algebras
is explored for both first order integrals and those of higher orders. Again
the peculiar position of second order equations is revealed.
In the seventh chapter the generalised Emden-Fowler equation is given a
more modern and complete treatment.
The final chapter looks at one of the fundamental algebras associated with
ordinary differential equations, the three element 8£(2, R), which is found in all
higher order equations of maximal symmetry, is a fundamental feature of the
Pinney equation which has played so prominent a role in the study of nonautonomous
Hamiltonian systems in Physics and is the signature of Ermakov
systems and their generalisations. / Thesis (Ph.D.)-University of Natal, 1995.
|
Page generated in 0.0481 seconds