This paper is concerned with the relationships between certain covering systems useful for set differentiation and with their application to density theorems and approximate continuity. The covering systems considered are the Vitali systems (which we call V-systems), the systems introduced by Sion (which we call S-systems), and a modification of the tile systems (which we call T-systems). It is easily checked from the definitions that V-systems are S-systems, and under slight restrictions, T-systems. We show also that under certain conditions S-systems are T-systems, and that in general the converses do not hold. Density theorems and the relationships between approximate continuity and measurability of functions are discussed for these systems. In particular, we prove that for T-systems measurable functions are approximately continuous and hence a density theorem holds. / Science, Faculty of / Mathematics, Department of / Graduate

Families of three- and four-point corrector formulae are derived, which differ from standard formulae in that they express yո in terms of more than one previously computed ordinate. It is shown that the standard formulae are special cases of the more general formulae derived here. By theoretical argument and by numerical experiments it is shown that the standard formulae are often inferior to others which are developed in this thesis. The three-point family, with its associated truncation error, is given in (7) and (9) of Chapter 2 on page 12. The four-point family is given in (41) on page 25. With the help of Rutishauser's method each family is examined for stability. In the four-point case a procedure is described, whereby the magnitude of the coefficient in the error term can be minimized subject to the restriction that the formula shall remain stable. Also a theorem is proved, which states that no stable four-point formula can have a truncation error of degree higher than fifth in the step-size h. / Science, Faculty of / Mathematics, Department of / Graduate

After a brief discussion of numerical methods for the solution of the ordinary differential equation x'= f(t, x) the problem of finding optimum methods is considered. The thesis then deals with this problem for the family of Runge-Kutta methods. Criteria for optimum methods are discussed and then the derivation of third-order methods is examined in detail. The next part of the thesis deals with possible approaches to finding optimum methods. The first approach consists of finding some sort of estimate for f and its derivatives contained in the truncation error T[subscript n]. The resulting expression is then dependent on some free parameter or parameters (depending on the order of the method) which are chosen in order to minimize this expression. The second approach assumes the independence of terms in the truncation error and minimizes, in some sense, the coefficients of these terms. Several procedures based on these approaches, are used to predict optimum second-order and third-order methods and comparisons are made with experimental results. While no definite conclusions could be drawn it was seen that one particular procedure gave a good prediction. This result encourages further studies in this area. I hereby certify that this abstract is satisfactory. / Science, Faculty of / Mathematics, Department of / Graduate

The theory of Poincaré and Bendixson is applied to establish the existence of periodic solutions of the differential equation ẍ + f(x, ẋ)ẋ + g(x) x = 0 One part of the work is concerned with those equations which can be considered as arising from small perturbations of other equations of the same type, already possessing periodic solutions. Two existence theorems are demonstrated and the stability and uniqueness of the periodic solutions is also discussed. The other part contains several, theorems stating sufficient conditions for existence of periodic solutions which cannot be treated by perturbation methods. / Science, Faculty of / Mathematics, Department of / Graduate

A method is given by which a-differential equation with initial conditions can be converted into an integral equation. This procedure is used to derive the Multiplication Theorems for Bessel functions, and to obtain an expansion of the confluent hyper geometric function in terms, of Bessel functions. The method is adapted to find approximate eigenvalues: and eigenfunctions of bounded quantum mechanical problems, and to obtain an approximate solution of a non-linear differential equation. / Science, Faculty of / Mathematics, Department of / Graduate

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Dissertation (Ph.D.)--Auburn University,2007. / Abstract. Includes bibliographic references (p.99-103).