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The thesis presents the results of our research on symmetry for some semilinear elliptic problems and on existence of solution for quasilinear problems involving singularities. The text is composed of two parts, each of which begins with a specific introduction. The first part is devoted to symmetry and symmetry-breaking results. We study a class of partial differential equations involving radial weights on balls, annuli or $R^N$ --where these weights are unbounded--. We show in particular that on unbounded domains, focusing on symmetric functions permits to recover compactness, which implies existence of solutions. Then, we stress the fact that symmetry-breaking occurs on bounded domains, depending both on the weights and on the nonlinearity of the equation. We also show that for the considered class of problems, the multibumps-solution phenomenon appears on the annulus as well as on the ball. The second part of the thesis is devoted to partial and ordinary differential equations with singularities. Using concentration-compactness tools, we show that a rather large class of functionals is lower semi-continuous, leading to the existence of a ground state solution. We also focus on the unicity of solutions for such a class of problems.

We consider bounded weak solutions u of a degenerate parabolic-hyperbolic equation defined in a subset [mathematical symbols]. We define strong notion of trace at the boundary [mathematical symbols] reached by L¹ convergence for a large class of functionals of u. Such functionals depend on the flux function of the degenerate parabolic-hyperbolic equation and on the boundary. We also prove the well-posedness of the entropy solution for scalar conservation laws with a strong boundary condition with the above trace result as applications. / text

In this thesis, Newman-Penrose techniques are used to obtain some new exact solutions to Einstein's field equations of general relativity and to assist in the physical interpretation of some exact radiative space-times. Attention is restricted to algebraically special space-times with a twist-free, repeated principal null congruence. In particular, the Robinson-Trautman type N solutions, which describe expanding gravitational waves, are investigated for all possible values of the cosmological constant A and the Gaussian curvature parameter E. The wave surfaces are always (hemi-)spherical, with successive surfaces displaced along time-like, space-like or null lines, depending on E. Explicit sandwich waves of this class are studied in Minkowski, de Sitter or anti-de Sitter backgrounds and a particular family of such solutions, which can be used to represent snapping or decaying cosmic strings, is considered in detail. The singularity and global structure of the solutions is also presented. In the remaining part of the thesis, the complete family of space-times with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves), that are of algebraic type III and for which the cosmological constant (Ac) is non-zero, is presented. The possible presence of an aligned pure radiation field is also assumed. These space-times generalise the known vacuum solutions of type N with arbitrary Ac and type III with Ac = O. It is shown that there are two, one and three distinct classes of solutions when Ac is respectively zero, positive and negative and, in these cases, the wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively. The singularities which occur in these space-times are interpreted in terms of envelopes of these wave surfaces. Again, by considering functions of the retarded time which "cross-over" between canonical types, sandwich waves are also studied. The limiting cases of these, giving rise to shock or impulsive waves, are also considered.

LD2668 .T4 1942 F3 / Master of Science

Call number: LD2668 .T4 1978 H47 / Master of Science

Submitted in fulfilment of the academic requirements for the Masters Degree in Technology: Chemistry, Durban University of Technology,2013. / The thermodynamic properties of binary liquid mixtures using an ionic liquid (IL) with alcohols were determined at different temperatures. The ionic liquid used was 1-butyl-3- methylimidazolium methylsulphate [BMIM]+[MeSO4]-. Densities, speed of sound, and refractive indices for the binary mixtures ([BMIM]+[MeSO4]- + methanol, or 1-propanol, or 2-propanol, or 1-butanol) were experimentally measured over the whole range of composition at T = (298.15, E 303.15, 308.15, and 313.15) K. From the experimental data, excess molar volumes, V m , E , deviations in refractive isentropic compressibilities, κ s , excess isentropic compressibilities, κ S indices, ∆n, and molar refractions, R, were calculated. The excess partial molar volumes were also calculated at T = 298.15 K. For the binary systems, ([BMIM]+[MeSO4]- + methanol, or 1-propanol, or 2-propanol, or E E E 1-butanol) V m and κ S are always negative and V m decrease slightly when the temperature increases. The refractive index deviation at T = (298.15, 303.15, 308.15, and 313.15) K is positive over the whole composition range. The measured negative values for excess molar volume of these mixtures ([BMIM]+[MeSO4]- + methanol, or 1-propanol, or 2-propanol, or 1-butanol) indicate strong ion-dipole interactions and packing between alcohols and IL are present. The Redlich-Kister smoothing polynomial equation was satisfactorily applied for the E E fitting of the V m , κ S , and ∆n data to give the fitting parameters and the root-mean-square deviations. The Lorentz-Lorenz (L-L) equation was also used to correlate the volumetric property and predict the density or refractive index of the binary mixtures of ionic liquid and the organic solvents. The Lorentz-Lorenz approximation gives a higher σ when used to correlate the iiiexcess molar volumes for the mixtures ([BMIM]+[MeSO4]- + methanol, or 1-propanol, or 2-propanol, or 1-butanol). The L-L equation gives good results for the prediction of density and refractive index. The results are discussed in terms of solute-solute, solute-solvent and solvent- solvent interactions. / National Research Foundation

Numerical methods for solving initial value problems in ordinary differential equations are studied. A notation is introduced to represent cyclic methods in terms of two matrices, A_h, and B_h, and this is developed to cover the very extensive class of m-block methods. Some stability results are obtained and convergence is analysed by means of a new consistency concept, namely optimal consistency. It is shown that optimal consistency allows one to give two-sided bounds on the global error, and examples are given to illustrate this. The form of the inverse of A_h is studied closely to give a criterion for the order of convergence to exceed that of consistency by one. Further convergence results are obtained , the first of which gives the orders of convergence for cases in which A_h, and B_h, have a special form, and the second of which gives rise to the possibility of the order of convergence exceeding that of consistency by two or more at some stages. In addition an alternative proof is given of the superconvergence result for collocation methods. In conclusion the work covered is set in the context of that done in recent years by various authors.