The derivation of centered compact schemes at interior and boundary grid points is performed and an analysis of stability and computational efficiency is given. Compact schemes are high order implicit methods for numerical solutions of initial and/or boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique is employed. Experiments for non-stationary linear problems (convection, heat conduction) and also for nonlinear problems (Burgers' and KdV equations) were performed. The compact solvers were combined with Euler and fourth-order Runge-Kutta time differencing. In most cases, the order of convergence of the numerical solution to the exact solution was the same as the formal order of accuracy of the compact schemes employed.

In this thesis I will outline the impact of Pell's equation on various branches of number theory, as well as some of the history. I will also discuss some recently discovered properties of the solutions of Pell's equation.

Yes / The prevalence of micromoulded components has steadily increased over recent years. The production of such components is extremely sensitive to a number of variables that may potentially lead to significant changes in the surface geometry, often regarded as a crucial determinant of the product¿s functionality and quality. So far, traditional large-scale quality assessment techniques have been used in micromoulding. However, these techniques are not entirely suitable for small scales . Techniques such as Atomic Force Mi- croscopy (AFM) or White Light Interferometry (WLI) have been used for obtaining full three-dimensional profiles of micromoulded components, pro- ducing large data sets that are very difficult to manage. This work presents a method of characterizing surface features of micro and nano scale based on the use of the Biharmonic equation as means of describing surface profiles whilst guaranteeing tangential (C1) continuity. Thus, the problem of rep- resenting surface features of micromoulded components from massive point clouds is transformed into a boundary-value problem, reducing the amount of data required to describe any given surface feature.The boundary conditions needed for finding a particular solution to the Biharmonic equation are extracted from the data set and the coefficients associated with a suitable analytic solution are used to describe key design parameters or geometric properties of a surface feature. Moreover, the expressions found for describ- ing key design parameters in terms of the analytic solution to the Biharmonic equation may lead to a more suitable quality assessment technique for mi- cromoulding than the criteria currently used. In summary this technique provides a means for compressing point clouds representing surface features whilst providing an analytic description of such features. The work is applicable to many other instances where surface topography is in need of efficient representation.

No / Polypropylene sheets have been stretched at 160 °C to a state of large biaxial strain of extension ratio 3, and the stresses then allowed to relax at constant strain. The state of strain is reached via a path consisting of two sequential planar extensions, the second perpendicular to the first, under plane stress conditions with zero stress acting normal to the sheet. This strain path is highly relevant to solid phase deformation processes such as stretch blow moulding and thermoforming, and also reveals fundamental aspects of the flow rule required in the constitutive behaviour of the material. The rate of decay of stress is rapid, and such as to be highly significant in the modelling of processes that include stages of constant strain. A constitutive equation is developed that includes Eyring processes to model both the stress relaxation and strain rate dependence of the stress. The axial and transverse stresses observed during loading show that the use of a conventional Levy-Mises flow rule is ineffective, and instead a flow rule is used that takes account of the anisotropic state of the material via a power law function of the principal extension ratios. Finally the constitutive model is demonstrated to give quantitatively useful representation of the stresses both in loading and in stress relaxation.

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In this paper we examine the effectiveness of reducing the second order radial equation, of the hyperspherical coordinate solution to the two-electron Schrodinger equation, into a set of coupled first order linear equations as suggested by Klar. All results have been obtained in a completely nonadiabatic formalism thereby ensuring accuracy. We arrive at the conclusion that our application of the reduction process is in some way inconsistent and suggest a possible resolution to this anomaly.

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