Thesis--Friedrich-Alexanders-Universität Erlangen. / Lebenslauf. Bibliographical footnotes.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 56-57).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 72-75).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1964. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

In this work I study certain aspects of qualitative behaviour of solutions to nonlinear PDEs. The thesis consists of introduction and three parts. In the first part I study solutions of Emden-Fowler type elliptic equations in nondivergence form. In this part I establish the following results; 1. Asymptotic representation of solutions in conical domains; 2. A priori estimates for solutions to equations with weighted absorption term; 3. Existence and nonexistence of positive solutions to equations with source term in conical domains. In the second part I study regularity properties of nonlinear degenerate parabolic equations. There are two results here: A Harnack inequality and the H51der continuity for solutions of weighted degenerate parabolic equations with a time-independent weight from a suitable Muckenhoupt class; A new proof of the Holder continuity of solutions. The third part is propedeutic. In this part I gathered some facts and simple proofs relating to the Harnack inequality for elliptic equations. Both divergent and nondivergent case are considered. The material of this chapter is not new, but it is not very easy to find it in the literature. This chapter is built entirely upon the so-called "growth lemma" ideology (introduced by E.M. Landis).

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

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

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 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

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