M.Sc. / A class of numerical methods for solving nonstiff initial value problems in ordinary differential equations has been developed. These methods, designated RKrGLn, are based on a Runge-Kutta method of order r (RKr), and Gauss-Legendre integration over n+ 1 nodes. The interval of integration for the initial value problem is subdivided into an integer number of subintervals. On each of these n + 1 nodes are defined in accordance with the zeros of the Legendre polynomial of degree n. The Runge-Kutta method is used to find an approximate solution at each of these nodes; Gauss-Legendre integration is used to find the solution at the endpoint of the subinterval. The process then carries over to the next subinterval. We find that for a suitable choice of n, the order of the local error of the Runge- Kutta method (r + 1) is preserved in the global error of RKrGLn. However, a poor choice of n can actually limit the order of RKrGLn, irrespective of the choice of r. What is more, the inclusion of Gauss-Legendre integration slightly reduces the number of arithmetical operations required to find a solution, in comparison with RKr at the same number of nodes. These two factors combine to ensure that RKrGLn is considerably more efficient than RKr, particularly when very accurate solutions are sought. Attempts to control the error in RKrGLn have been made. The local error has been successfully controlled using a variable stepsize strategy, similar to that generally used in RK methods. The difference lies in that it is the size of each subinterval that is controlled in RKrGLn, rather than each individual stepsize. Nevertheless, local error has been successfully controlled for relative tolerances ranging from 10 -4 to 10-10 . We have also developed algorithms for estimating and controlling the global error. These algorithms require that a complete solution be obtained for a specified distribution of nodes, after which the global error is estimated and then, if necessary, a new node distribution is determined and another solution obtained. The algorithms are based on Richardson extrapolation and the use of low-order and high-order pairs. The algorithms have successfully achieved desired relative global errors as small as 10-1° . We have briefly studied how RKrGLn may be used to solve stiff systems. We have determined the intervals of stability for several RKrGLn methods on the real line, and used this to develop an algorithm to solve a stiff problem. The algorithm is based on the idea of stepsize/subinterval adjustment, and has been used to successfully solve the van der Pol system. Lagrange interpolation on each subinterval has been implemented to obtain a piecewise continuous polynomial approximation to the numerical solution, with same order error, which can be used to find the solution at arbitrary nodes.

No description available.

No description available.

The object of this dissertation is to discuss the stability in the large of the trivial solution for systems of two differential equations using qualitative methods (of course in combination with the construction of Lyapunov function). The right hand sides of these systems do not contain the time t explicitly. First of all we discuss (Sec. 2.) the system of the type [equations omitted] These equations occur in automatic regulation. Using qualitative methods we determine sufficient conditions in order that the trivial solution of system (l) be asymptotically stable in the large. In this connection we note that a theorem proved by Aĭzerman for the systems of two equations (Sec. 3), namely, for the systems [equations omitted] In the case of system (2) we give a new proof of a theorem which asserts that if c² + ab ≠ o, then the trivial solution is asymptotically stable in the large under the generalized Hurwitz conditions. The theorem was first proved by Erugin [8]. For system (3) Malkin showed that the trivial solution is asymptotically stable in the large under the conditions a + c < o, (acy - bf(y)) y > o for y ≠ o and [formula omitted] We prove a similar theorem without the requirement of [formula omitted] We also discuss (Sec. 4) the stability in the large of the systems [equations omitted] We consider (Sec. 5) again the system of the type (l) but under assumptions as indicated by Ershov [6] who has discussed various cases where the asymptotic stability in the large holds. Not agreeing fully with the proofs of these theorems we give our own proofs. Finally we discuss (Sec. 6 and 7) the stability in the large of the systems [equations omitted] under suitable assumptions. As a sample case we prove that if ab > o,then the trivial solution of system (4) is asymptotically stable in the large under conditions h₁(y) + h₂(x) < o , h₁(y) h₂(x) - ab > o, for x ≠ o, y ≠ o / Science, Faculty of / Mathematics, Department of / Graduate

A method of writing the solution of a second order differential equation through a Volterra Integral Equation is developed. The method is applied to initial value problems, to special functions, and to bounded Quantum Mechanical problems. Some of the results obtained are original, and other results agree essentially with the work done previously by others. / Science, Faculty of / Mathematics, Department of / Graduate

In the first part of this thesis, strong comparison theorems of Sturm's type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to non-self-adjoint quasilinear systems. Application of these results to first order systems leads to criteria for the existence of conjugate points. In the second part, comparison theorems are obtained for elliptic differential operators of arbitrary even order. A description of the behaviour of the smallest eigenvalue for such operators is given under domain perturbations by means of Garding's inequality. New oscillation and nonoscillation criteria are obtained by variational methods. Specialization of our theorems to elliptic equations of fourth order, and to ordinary differential equations yields various generalizations of known results. / Science, Faculty of / Mathematics, Department of / Graduate

A method is presented for determining approximate solutions to a class of grossly nonlinear, non-autonomous second order differential equations characterized by [formula omitted] with the restriction that resonance effects be negligible. Solutions are developed in terms of the Jacobian elliptic functions, and may be related directly to the degree, of non-linearity in the differential equation. An integral error definition, which can be applied to any particular differential equation, is used to portray regions of validity of the approximate solution in terms of equation parameters. In practice the approximate solution is shown to be of greater accuracy than would be expected from the error analysis, and use of the error diagram leads to a pessimistic estimate of solution accuracy. Two autonomous equations are considered to facilitate comparison between the elliptic function approximation and that obtained from the method of Kryloff and Bogoliuboff. The elliptic function solution is shown to be accurate even for heavily damped nonlinear autonomous equations, when the quasi-linear approximation of Kyrloff-Bogoliuboff cannot with validity be applied. Four examples are chosen, from the fields of astrophysics, mechanics, circuit theory and control systems to illustrate, some areas to which the general approximation method relates. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Thesis Supervisor: C. A. Swanson. New comparison and Sturm-type theorems are established which enable us to extend known oscillation and non-oscillation criteria to: (1) non-self-adjoint operators, (2) quasi-linear operators, (3) fourth order operators of a type not previously-considered. Since the classical principle of Courant does not hold for some of the operators considered, the comparison theorems involve, in part, new estimates on the location of the smallest eigenvalue of the operators in question. A description of the behaviour of the eigenvalue as the domain is perturbed is also given for such operators by the use of Schauder's "a priori" estimates. The Sturm-type theorems are proved by topological arguments and extended to quasi-linear as well as to non-self-adjoint operators. The fourth order operators considered are of a type which does not yield forms identical to those arising in second order problems. Some examples illustrating the theory are given. / Science, Faculty of / Mathematics, Department of / Graduate

Criteria will be obtained for a linear self-adjoint elliptic partial differential equation to be oscillatory or nonoscillatory in unbounded domains R of n-dimensional Euclidean space Eⁿ. The criteria are of two main types: (i) those involving integrals of suitable majorants of the coefficients, and (ii) those involving limits of these majorants as the argument tends to infinity. Our theorems constitute generalizations to partial differential equations of well-known criteria of Hille, Leighton, Potter, Moore, and Wintner for ordinary differential equations. In general, our method provides the means for extending in this manner any oscillation criterion for self-adjoint ordinary differential equations. Our results imply Glazman's theorems in the special case of the Schrodinger equation in Eⁿ. In the derivation of the oscillation criteria it is assumed that R is either quasiconical (i.e. contains an infinite cone) or limit-cylindrical (i.e. contains an infinite cylinder). In the derivation of the nonoscillation criteria no special assumptions regarding the shape of the domain are needed. Examples illustrating the theory are given. In particular, it is shown that the limit criteria obtained in the second order case are the best possible of their kind. / Science, Faculty of / Mathematics, Department of / Graduate

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