Mathematics and Computer Science Education

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Amplitude-shape method for the numerical solution of ordinary differential equations.
(1997) Parumasur, Nabendra.; Banasiak, Jacek.; Mika, Janusz R.
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.
Coagulation-fragmentation dynamics in size and position structured population models.
(2008) Noutchie, Suares Cloves Oukouomi.; Banasiak, Jacek.
One of the most interesting features of fragmentation models is a possibility to breach
Long time behaviour of population models.
(2010) Namayanja, Proscovia.; Banasiak, Jacek.; Willie, Robert.
Non-negative matrices arise naturally in population models. In this thesis, we look at the theory of such matrices and we study the Perron-Frobenius type theorems regarding their spectral properties. We use these theorems to investigate the asymptotic behaviour of solutions to continuous time problems arising in population biology. In particular, we provide a description of long-time behaviour of populations depending on the nature of the associated matrix. Finally, we describe a few applications to population biology.
A measure-theoretic approach to chaotic dynamical systems.
(2001) Singh, Pranitha.; Banasiak, Jacek.
The past few years have witnessed a growth in the study of the long-time behaviour of physical, biological and economic systems using measure-theoretic and probabilistic methods. In this dissertation we present a study of the evolution of dynamical systems that display various types of irregular behaviour for large times. Large systems, containing many elements, like e.g. bacteria populations or ensembles of gas particles, are very difficult to analyse and contain elements of uncertainty. Also, in general, it is not necessary to know the evolution of each bacteria or each gas particle. Therefore we replace the "pointwise" description of the evolution of the system with that of the evolution of suitable averages of the population like e.g. the gas or the bacteria spatial density. In particular cases, when the quantity in the evolution that we analyse has the probabilistic interpretation, say, the probability of finding the particle in certain state at certain time, we will be talking about the evolution of (probability) densities. We begin with the establishment of results for discrete time systems and this is later followed with analogous results for continuous time systems. We observe that in many cases the system has two important properties: at each step it is determined by a non-negative function (for example the spatial density or the probability density) and the overall quantity of the elements remains preserved. Because of these properties the most suitable framework to investigate such systems is the theory of Markov operators. We shall discuss three levels of "chaotic" behaviour that are known as ergodicity, mixing and exactness. They can be described as follows: ergodicity means that the only invariant sets are trivial, mixing means that for any set A the sequence of sets S-n(A) becomes, asymptotically, independent of any other set B, and exactness implies that if we start with any set of positive measure, then, after a long time the points of this set will spread and completely fill the state space. In this dissertation we describe an application of two operators related to the generating Markov operator to study and characterize the abovementioned properties of the evolution system. However, a system may also display regular behaviour. We refer to this as the asymptotic stability of the Markov operator generating this system and we provide some criteria characterizing this property. Finally, we demonstrate the use of the above theory by applying it to a system that is modeled by the linear Boltzmann equation.
A new approach to ill-posed evolution equations : C-regularized and B- bounded semigroups.
(2001) Singh, Virath Sewnath.; Banasiak, Jacek.
The theory of semigroups of linear operators forms an integral part of Functional Analysis with substantial applications to many fields of the natural sciences. In this study we are concerned with the application to equations of mathematical physics. The theory of semigroups of bounded linear operators is closely related to the solvability of evolution equations in Banach spaces that model time dependent processes in nature. Well-posed evolution problems give rise to a semigroup of bounded linear operators. However, in many important and interesting cases the problem is ill-posed making it inaccessible to the classical semigroup theory. One way of dealing with this problem is to generalize the theory of semigroups. In this thesis we give an outline of the theory of two such generalizations, namely, C-regularized semigroups and B-bounded semigroups, with the inter-relations between them and show a number of applications to ill-posed problems.

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