9 research outputs found
The Finite Spectrum of Fourth-Order Boundary Value Problems with Transmission Conditions
A class of fourth-order boundary value problems with transmission conditions are investigated. By constructing we prove that these class of fourth order problems consist of finite number of eigenvalues. Further, we show that the number of eigenvalues depend on the order of the equation, partition of
the domain interval, and the boundary conditions (including the transmission conditions) given
Forward and inverse spectral theory of Sturm-Liouville operators with transmission conditions
Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, School of Mathematics, 2017.ForwardandinversespectralproblemsconcerningSturm-Liouvilleoperatorswithoutdiscontinuitieshavebeenstudiedextensively. Bycomparison,therehasbeenlimitedworktacklingthecase where the eigenfunctions have discontinuities at interior points, a case which appears naturally in physical applications. We refer to such discontinuity conditions as transmission conditions. We consider Sturm-Liouville problems with transmission conditions rationally dependent on the spectral parameter. We show that our problem admits geometrically double eigenvalues, necessitating a new analysis. We develop the forward theory associated with this problem and also consider a related inverse problem. In particular, we prove a uniqueness result analogous to that of H. Hochstadt on the determination of the potential from two sequences of eigenvalues. In addition, we consider the problem of extending Sturm’s oscillation theorem, regarding the number of zeroes of eigenfunctions, from the classical setting to discontinuous problems with general constant coefficient transmission conditionsGR201
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd
Spectral theory of self-adjoint higher order differential operators with eigenvalue parameter dependent boundary conditions
We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue
parameter dependent boundary conditions and secondly a sixth-order di erential operator
with eigenvalue parameter dependent boundary conditions. We associate to each of these
problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral
proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst
four terms of the eigenvalue asymptotics