This thesis is a study of stability of the solutions of differential equations.

It is the object of this paper to present a method for finding the roots of any algebraic equation.

We study positive solutions to classes of steady state reaction diffusion equations that arise naturally in applications. In particular, we study models arising from population dynamics and combustion theory. The main focus of this dissertation is the mathematical analysis of a challenging new class of problems when a certain nonlinear boundary condition is satisfied. In particular, we establish existence and multiplicity results by making use of the Quadrature method, the method of sub-super solutions, and degree theory. The results in this dissertation provide a significant contribution towards the analysis of elliptic boundary value problems with nonlinear boundary conditions.

In this thesis, we will study basic mathematical epidemic models SIR, SIS, and SEIR. Then we will construct a modified model as a combination of SIR and SIS models. First, we will find the explicit solutions for the SIS model. and show no exact solution for the SIR model. Also, find the parametric solution for the SIR model and find a numerical solution by using Euler’s method. Then we find an approximate explicit form of the epidemic curve. Also, we will study parametric influences on the SIR and SIS models. Finally, we will suggest some recommendations to decrease the epidemic’s spread.

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

We study positive solutions to classes of nonlinear elliptic singular problems of the form: -Δpu = λ g(u) uα in Ω u = 0 on δΩ where Ω is a bounded domain in ℝN, N ≥ 1 with smooth boundary δΩ, &lambda¸ is a positive parameter, α ∈(0; 1), Δpu := div(⌊∇u⌋p-2 ∇u); p > 1 is the p-Laplacian operator, and g is a smooth function. Such elliptic problems naturally arise in the study of steady state reaction diffusion processes. In particular, we will be interested in the challenging new class of problems when g(0) < 0 (hence lims→0+g(s) sα = - ∞ which we refer to as infinite semipositone problems. Our focus is on existence results. We obtain results for the single equation case as well as to the case of systems. We use the method of sub-super solutions to prove our results. The results in this dissertation provide a solid foundation for the analysis of such infinite semipositone problems.

No description available.

Thesis (MSc)--Stellenbosch University, 2004 / ENGLISH ABSTRACT: In an ongoing study an empirical force field that can correctly model N,N'- dialkylimidazolium halide ionic liquids, the Imidazolium Ionic Liquid Force Field - IILFF, was developed based on experimental data obtained from the Cambridge Structural Database (CSD) and data calculated using Gaussian98. Different conformations of the isolated cations were optimised at the Hartree-Fock level using the 6-31G(d) basis set. Structural, vibrational and partial atomic charge data of the lowest energy conformation of each cation were taken as observables during optimisation of the force field parameters. Initial parameters of the IILFF were taken from existing force fields and were optimised using the above mentioned data. The IILFF was used to minimise isolated cations as well as crystals. These results were successfully tested against isolated cations minimised using Gaussian98 and the experimental crystals. Finally, the melting process of the 1,3-dimethylimidazolium chloride crystal was studied using an NPT ensemble starting from an ordered crystal cell and increasing the simulation temperature beyond the experimental melting temperature. The IILFF was then used to calculate the potential energy of the system. / AFRIKAANSE OPSOMMING: In 'n voortgaande studie om 'n empiriese kragveld te bou wat N,N' -dialkielimidasoliumhalied ioniese vloeistowwe korrek kan modelleer is die Imidasolium Ioniese Vloeistof Kragveld (nVK) ontwikkel. Die kragveld is ontwikkel gebasseer op eksperimentele data verkry uit die Cambridge Strukturele Databasis (CSD) asook uit data vanaf Gaussian98 berekeninge. Verskillende konformasies van die geïsoleerde katione is geoptimiseer deur middel van Hartree Fock 6-31G(d) berekeninge. Strukturele data, asook vibrasies en gedeeltelike atoom ladings van die laagste energie konformasie van elke katioon is gebruik as waarneembare veranderlikes vir die bepaling van die optimale kragveld parameters. Beginwaardes vir die nVK is geneem uit bestaande kragvelde en geoptimiseer met behulp van bogenoemde data. Die IIVK is gebruik om geïsoleerde katione asook kristalle te minimiseer. Die resultate is suksesvol getoets teen geïsoleerde katione wat met behulp van Gaussian98 geminimiseer is en eksperimenteel bepaalde kristalle. Laastens is die smeltproses van die 1,3-dimetielimidasolium chloried kristal bestudeer met behulp van 'n NPT ensemble. Daar is begin by 'n geordende kristal en die simulasie temperatuur is verhoog tot meer as die eksperimentele smeltpunt. Die IIVK is dan gebruik om die potensiële energie van die sisteem te bepaal.

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy