The subject matter of this thesis consists of a qualitative study of the stability and asymptotic stability of the zero solution of certain types of nonlinear differential equations, for arbitrary initial perturbations, and the construction of a periodic solution for a Hamiltonian system with n( ≥ 2) degrees of freedom. The material is divided into three chapters. The stability of the system (1) ẋ = xh₁(y) + ay, ẏ = f(x) + yh₂(x) with some restrictions on the functions h₁ (y), h₂(x) and f(x), is discussed in the first chapter. It turns out that some of the results proved by I.H. MUFTI ([l], [2], [3]), for the systems (2) ẋ = xh₁(y) + ay, ẏ = xh₂(x) + by and (3) ẋ = xh₁(y) + ay, ẏ = bx + yh₂(x) become particular cases of our results for system (1). Consequently an answer in the affirmative has been given to a problem proposed by I.H. MUFTI [1]. In the same chapter a generalization to the problem of M. A. AIZERMAN [l] for the case n = 2 is given in the form (4) ẋ = f₁(x) + f₂(y), ẏ = ax + f₃(y). This system has been discussed first by a qualitative method and second by constructing a LYAPUNOV function. In chapter II, stability of a quasilinear equation (5) [formula omitted] is discussed, by using LYAPUNOV's second method. It has been proved that if (i) [formula omitted] (ii) [formula omitted] for all values of x and y = ẋ (iii) [formula omitted] for all x,y (iv) [formula omitted] (where G,g and w are defined in Theorem 2.1) (v) [formula omitted] then the zero solution of (5) is asymptotically stable for arbitrary initial perturbations. In the same chapter certain equations of third order have also been discussed for "complete stability". These equations are special cases of (5) and are more general than those considered by SHIMANOV [l] and BARBASHIN [l]. AIZERMAN's [l] problem for the case n = 3 is generalized to two different forms, one of which is (6) [formula omitted] which is more general than the forms considered by V.A. PLISS [4] and N.N. KRASOVSKII [l]. Under a non-singular linear transformation equations(6) assume the form (7) [formula omitted] It has been proved that if in addition to the usual existence and uniqueness requirements, the conditions (i) [formula omitted] (ii) [formula omitted] (iii) [formula omitted] are fulfilled, then the zero solution of (7) is asymptotically stable in the large. In the third chapter a Hamiltonian system with n (≥ 2) degrees of freedom is considered in the normalized form (8)[formula omitted] where fĸ are power series in zk beginning with quadratic terms. A periodic solution for system (8) is constructed in the form (9) [formula omitted] where [formula omitted] is a homogeneous polynomial of degree [formula omitted] in terms of four time dependent variables a, B, y, õ. C. L. SIEGEL [l] constructs a periodic solution in terms of two variables [formula omitted] under the assumption that the corresponding linear system has a pair of purely imaginary eigenvalues. Here it is assumed that the linear system possesses two distinct pairs of purely imaginary eigenvalues and this necessitates the consideration of four time dependent variables in the construction of the periodic solution. / Science, Faculty of / Mathematics, Department of / Graduate

A continuous Markov process is examined for the purpose of developing Monte Carlo methods for solving partial differential equations. Backward Kolmogorov equations for conditional probability density functions and more general equations satisfied by auxiliary probability density functions are derived. From these equations and the initial and boundary conditions that the density functions satisfy, it is shown that solutions of partial differential equations at an interior point of a region can be written as the expected value of randomly-selected initial and boundary values. From these results, Monte Carlo methods for solving homogeneous and nonhomogeneous elliptic, and homogeneous parabolic partial differential equations are proposed. Hybrid computer techniques for mechanizing the Monte Carlo methods are given. The Markov process is simulated on the analog computer and the digital computer is used to control the analog computer and to form the required averages. Methods for detecting the boundaries of regions using analog function generators and electronic comparators are proposed. Monte Carlo solutions are obtained on a hybrid system consisting of a PACE 231 R-V analog computer and an ALWAC III-E digital computer. The interface for the two computers and a multichannel discrete-interval binary-noise source are described. With this equipment, solutions having a small variance are obtained at a rate of approximately five minutes per solution. Example solutions are given for Laplace's equation in two and three dimensions, Poisson's equation in two dimensions and the heat equation in one, two and three dimensions. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In the study of many engineering systems involving nonlinear elements such as a saturating inductor in an electrical circuit or a hard spring in a mechanical system, we face the problem of solving the equation ẍ + 2εẋ + x + μx³ = 0 which does not have an exact analytical solution,. Because a consistent framework is desirable in the course of the study, we can assume that the initial conditions are x(0) = 1 and ẋ(0) = 0 without loss of generality. This equation is studied in detail by using numerical solutions obtained from a digital computer. When ε and μ are small, classical methods such as the method of variation of parameters and averaging methods based on residuals provide analytical approximations to the equation and enable the engineer to gain useful insight into the system. However, when ε and μ are not small, these classical methods fail to yield acceptable results because they are all based on the assumption that the equation is quasi-linear. Therefore, two new analytical methods, namely: the parabolic phase approximation and the correction term approximation, are developed according to whether ε < 1 or ε ≥1, and are proven to be applicable for values of ε and μ far beyond the limit of classical methods. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

A modal finite element method is presented for the steady state and transient analyses of the plane flow of incompressible Newtonian fluid. The governing restricted functional is discretized with a high precision triangular stream function finite element. Eigenvalue analysis is carried out on the resulting discretized problem, under the assumption that the nonlinear convective term is equal to zero. After truncating at various levels of approximation to obtain a reduced number of modes, the transformation to the new vector space, spanned by these modes is performed. Advantage is taken of the ..symmetric and the antisymmetric properties of the modes in order to simplify the calculations. The Lagrange multipliers technique is employed to {incorporate the nonhomo-geneous boundary conditions. The steady state analysis is carried out by utilizing the Newton-Raphson iterative procedure. The algorithm for transient analysis is based upon backward finite differences in time. Numerical results are presented for the fully developed plane Poiseuille flow, the flow in a square cavity, and the flow over a circular cylinder problems. These resultscfor the steady state are compared with the results obtained by direct finite element approach on the same grids and the results obtained by finite differences technique. It is concluded that the number of modes, which are to be retained in the analysis in order to achieve reasonable results, increases with the refinement of the finite element grid. Furthermore, the choice of modes to be used depends on the problem. Finally it is established, that this new modal method yields good results in the range of moderate Reynolds numbers with about 50% or less of the modes of the problem. This, in turn, means that the time integrations can be performed on a greatly reduced number of equations and hence potential savings in computer time are significant. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

A necessary and sufficient condition is established for the existence of an invertible mapping of a system of nonlinear differential equations to a system of linear differential equations based on a group analysis of differential equations. It is shown how to construct the mapping, when it exists, from the invariance group of the nonlinear system. It is demonstrated that the hodograph transformation, the Legendre transformation and Lie's transformation of the Monge-Ampere equation are obtained from this theorem. The equation (ux)Puxx-uyy=0 is studied and it is determined for what values of p this equation is transformable to a linear equation by an invertible mapping. Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance groups of equations associated with the given nonlinear equations. A number of such; examples are given, including Burgers' equation uxx +uuz-ut=0 a nonlinear diffusion equation (u⁻²ux ) x -ut =0, equations of wave propagation {Vy-wx=0, Vy-avw-bv-cw=0}, equations of a fluid flow {wy+vx=0, wx -v⁻¹wP=0} and the Liouville equation uxy=eu. As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the invariance groups of the respective equations. All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations. / Science, Faculty of / Mathematics, Department of / Graduate

The research addresses the problem of maximizing a zero-one quadratic function. The report falls into three main sections. The first section uses results from Hammer [12] and Picard and Ratliff [23] to develop a new test for fixing the value of a variable in some solution and to provide a means for calculating a new upper bound on the maximum of the function. In addition the convergence of the method of calculation for the bounds is explored in an investigation of its sharpness. The second section proposes a branch and bound algorithm that uses the ideas of the first along with a heuristic solution procedure. It is shown that one advantage of this is that it may now be possible to identify how successful this algorithm will be in finding the maximum of a specified problem. The third section gives a basis for a new heuristic solution procedure. The method defines a concept of gradient which enables a simple steepest ascent technique to be used. It is shown that in general this will find a local maximum of the function. A further procedure to help distinguish between local and global maxima is also given. / Business, Sauder School of / Graduate

"This paper is a study of finite systems of homogenous linear inequalities, homogeneous linear equations, and nonhomogeneous equations. To each inequality or equation in one system there corresponds a nonnegative or unrestricted variable in the other and conversely. The array of coefficients in one system is the negative transpose in the other system. This duality furnishes the foundation for the duality of matrix games and linear programming"--Introduction. / Typescript. / "June, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaves 38-39).

"In 1824, the Norwegian mathematician N. H. Abel (1802-1829) proved that the general polynomial equation of degree greater than four with real numbers as coefficients is not solvable by radicals. That is, the roots cannot be expressed by a formula involving only rational operations and radicals. This result was unexpected, since formulas are known for the quadratic, cubic, and quartic equations. Another brilliant mathematician, E. Galois (1811-1832), used the concept of a group to penetrate further into the nature of polynomial equations. The object of this paper is to prove the insolvability of the quintic equation. In the process portions of the theory of field extensions and Galois theory are developed. Most of this material can be found in A Survey of Modern Algebra, by G. Birkhoff and S. MacLane. Certain questions, however, are treated in more detail than is found in most textbooks which contain the subject. This is especially true for the proof of the existence of a quintic equation not solvable by radicals"--Introduction. / "May 28, 1952." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Nickolas Heerema, Professor Directing Paper. / Includes bibliographical references (leaf 49).

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