Browsing by Author "Leach, Peter Gavin Lawrence."

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Algebraic properties of ordinary differential equations.
(1995) Leach, Peter Gavin Lawrence.
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.
Applications of symmetry analysis of partial differential and stochastic differential equations arising from mathematics of finance.
(2011) Nwobi, Felix Noyanim.; O'Hara, John Gerard.; Leach, Peter Gavin Lawrence.
In the standard modeling of the pricing of options and derivatives as generally understood these days the underlying process is taken to be a Wiener Process or a Levy Process. The stochastic process is modeled as a stochastic differential equation. From this equation a partial differential equation is obtained by application of the Feynman-Kac Theorem. The resulting partial differential equation is of Hamilton-Jacobi-Bellman type. Analysis of the partial differential equations arising from Mathematics of Finance using the methods of the Lie Theory of Continuous Groups has been performed over the last twenty years, but it is only in recent years that there has been a concerted effort to make full use of the Lie theory. We propose an extension of Mahomed and Leach's (1990) formula for the nth-prolongation of an nth-order ordinary differential equation to the nth-prolongation of the generator of an hyperbolic partial differential equation with p dependent and k independent variables. The symmetry analysis of this partial differential equation shows that the associated Lie algebra is {sl(2,R)⊕W₃}⊕s ∞A₁ with 12 optimal systems. A modeling approach based upon stochastic volatility for modeling prices in the deregulated Pennsylvania State Electricity market is adopted for application. We propose a dynamic linear model (DLM) in which switching structure for the measurement matrix is incorporated into a two-state Gaussian mixture/first-order autoregressive (AR (1)) configuration in a nonstationary independent process defined by time-varying probabilities. The estimates of maximum likelihood of the parameters from the "modified" Kalman filter showed a significant mean-reversion rate of 0.9363 which translates to a half-life price of electricity of nine months. Associated with this mean-reversion is the high measure of price volatility at 35%. Within the last decade there has been some work done upon the symmetries of stochastic differential equations. Here empirical results contradict earliest normality hypotheses on log-return series in favour of asymmetry of the probability distribution describing the process. Using the Akaike Information Criterion (AIC) and the Log-likelihood estimation (LLH) methods as selection criteria, the normal inverse Gaussian (NIG) outperformed four other candidate probability distributions among the class of Generalized Hyperbolic (GH) distributions in describing the heavy tails present in the process. Similarly, the Skewed Student's t (SSt) is the best fit for Bonny Crude Oil and Natural Gas log-returns. The observed volatility measures of these three commodity prices were examined. The Weibull distribution gives the best fit both electricity and crude oil data while the Gamma distribution is selected for natural gas data in the volatility profiles among the five candidate probability density functions (Normal, Lognormal, Gamma, Inverse Gamma and the Inverse Gaussian) considered.
Coherent structures and symmetry properties in nonlinear models used in theoretical physics.
(1994) Harin, Alexander O.; Leach, Peter Gavin Lawrence.; Barashenkov, I. V.
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.
Complete symmetry groups : a connection between some ordinary differential equations and partial differential equations.
(2008) Myeni, Senzosenkosi Mandlakayise.; Leach, Peter Gavin Lawrence.; O’Hara, J. G.
The concept of complete symmetry groups has been known for some time in applications to ordinary differential equations. In this Thesis we apply this concept to partial differential equations. For any 1+1 linear evolution equation of Lie’s type (Lie S (1881) Uber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung Archiv fur Mathematik og Naturvidenskab 6 328-368 (translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 473-508) containing three and five exceptional point symmetries and a nonlinear equation admitting a finite number of Lie point symmetries, the representation of the complete symmetry group has been found to be a six-dimensional algebra isomorphic to sl(2,R) s A3,1, where the second subalgebra is commonly known as the Heisenberg-Weyl algebra. More generally the number of symmetries required to specify any partial differential equations has been found to equal the number of independent variables of a general function on which symmetries are to be acted. In the absence of a sufficient number of point symmetries which are not solution symmetries one must look to generalized or nonlocal symmetries to remove the deficiency. This is true whether the evolution equation be linear or not. We report Ans¨ atze which provide a route to the determination of the required nonlocal symmetry or symmetries necessary to supplement the point symmetries for the complete specification of the equations. Furthermore we examine the connection of ordinary differential equations to partial differential equations through a common realisation of complete symmetry group. Lastly we revisit the notion of complete symmetry groups and further extend it so that it refers to those groups that uniquely specify classes of equations or systems. This is based on some recent developments pertaining to the properties and the behaviour of such groups in differential equations under the current definition, particularly their representations and realisations for Lie remarkable equations. The results seem to be quite astonishing.
Ermakov systems : a group theoretic approach.
(1993) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
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.
Extensions and generalisations of Lie analysis.
(1995) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations.
First integrals for the Bianchi universes : supplementation of the Noetherian integrals with first integrals obtained by using Lie symmetries.
(1997) Pantazi, Hara.; Leach, Peter Gavin Lawrence.
No abstract available.
Measurement and modelling of households' demand and access to basic water in relation to the rapidly increasing household numbers in South Africa.
(2010) Chidozie, Nnadozie Remigius.; Leach, Peter Gavin Lawrence.
Service delivery in post-apartheid South Africa has become a topical issue both in the academia and the political arena . The rise of social movements, the xenophobic tensions of May 2008 and protest actions could be noted as the major traits of post-apartheid South Africa. Though there are divergent views on the underlying causes of these protests, lack of service delivery has most significantly been at the centre stage. In this thesis we investigate the relationship between household/population changes and the demand for piped-water connection in South Africa. There is an ample, albeit at times of questionable accuracy, supply of statistics from official and other sources. These statistics are both the source of inspiration of particular societal measures to be investigated and a gauge of the accuracy of the mathematical/statistical modelling which is the central feature of this project. We construct mathematical/statistical models which take into account demographic constituents of the problem using differential equations for modelling household dynamics and we also investigate the interaction of demographic parameters and the demand for piped-water connection using multivariate statistical techniques. The results show that with a boost in delivery the rich provinces seem to be in better standing of meeting targets and that the increasing demand in household-based services could be most significantly attributed to the fragmentation of households against other demographic processes like natural increase in population and net migration. The results imply that in as much as service delivery policies and programmes should focus on formerly disadvantaged poor communities, adequate provisions for increasing service demands in urban centres should also be a priority in view of the increasing in-migration from rural areas as households fragment. Most of the findings/results are in tabular and graphical forms for easy understanding of the reader.
Noether's theorem and first integrals of ordinary differential equations.
(1997) Moyo, Sibusiso.; Leach, Peter Gavin Lawrence.
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.
The paradigms of mechanics : a symmetry based approach.
(1996) Lemmer, Ryan Lee.; Leach, Peter Gavin Lawrence.
An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics.
Singularity and symmetry analysis of differential sequences.
(2009) Maharaj, Adhir.; Leach, Peter Gavin Lawrence.; Euler, Marianna.
We introduce the notion of differential sequences generated by generators of sequences. We discuss the Riccati sequence in terms of symmetry analysis, singularity analysis and identification of the complete symmetry group for each member of the sequence. We provide their invariants and first integrals. We propose a generalisation of the Riccati sequence and investigate its properties in terms of singularity analysis. We find that the coefficients of the leading-order terms and the resonances obey certain structural rules. We also demonstrate the uniqueness of the Riccati sequence up to an equivalence class. We discuss the properties of the differential sequence based upon the equation ww''−2w12 = 0 in terms of symmetry and singularity analyses. The alternate sequence is also discussed. When we analyse the generalised equation ww'' − (1 − c)w12 = 0, we find that the symmetry properties of the generalised sequence are the same as for the original sequence and that the singularity properties are similar. Finally we discuss the Emden-Fowler sequence in terms of its singularity and symmetry properties.
Spherically symmetric charged Einstein-Maxwell solutions.
(1999) Pasha, Muhammad Akmal.; Maharaj, Sunil Dutt.; Leach, Peter Gavin Lawrence.
In this thesis we study spherically symmetric spacetimes with a perfect fluid source which incorporates charge. We seek explicit solutions to the Einstein- Maxwell system of equations. For nonaccelerating spherically symmetric models a charged, dust solution is found. With constant pressure the equations reduce to quadratures. Particular solutions are also found, with no acceleration, with the equation of state P =( y - 1)u. The Lie analysis is utilised to reduce the Einstein- Maxwell equations to a syst.em of ordinary differential equations. The evolution of the model depends on a Riccati equation for this general class of accelerating, expanding and shearing spacetimes with charge. Also arbitrary choices for the gravitational potentials lead to explicit solutions in particular cases. With constant gravitational potential A we generate a simple nonvacuum model. The analysis, in this case, enables us to reduce the solution to quadratures. With the value y = 2, for a stiff equation of state, we find that the solution is expressable in terms of elementary functions. Throughout the thesis we have attempted to relate our results to previously published work, and to obtain the uncharged perfect fluid limit where appropriate.