Bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm–Liouville problem

AbstractIn this paper we obtain bounds on the real and imaginary parts of non-real eigenvalues of a non-definite Sturm–Liouville problem, with Dirichlet boundary conditions, that improve on corresponding results obtained in Behrndt et al., [7]

On non-definite Sturm-Liouville problems with two turning points

This is an inaugural study of the Dirichlet problem associated with a regular non-definite Sturm-Liouville equation in the case of two turning points. We give a priori lower bounds on the Richardson numbers associated with this problem thereby complementing pioneering results by Atkinson and Jabon (1984) in the one turning point caseComment: 13 pages, To appear in Applied Math. and Computatio

On a non-definite Sturm-Liouville problem in the two turning point case - analysis and numerical results

In this paper we study the non-definite Sturm-Liouville problem comprising of a regular Sturm-Liouville equation and  Dirichlet boundary conditions on a closed interval. We consider the case in which the weight function changes sign twice in the given interval of definition. We give detailed numerical results on the spectrum of the problem, from which we verify various results on general non definite Sturm-liouville problems. We also present some theoretical results which support the numerical results. Some numerical results seem to be in contrast with the results that are so far obtained in the case where the weight function changes sign once. This leads to more open questions for future studies in this particular area

Some new results concerning general weighted regular Sturm-Liouville problems

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2

Qualitative and Spectral theory of some regular non-definite Sturm-Liouville problems

In this Licentiate thesis, we study some regular non-definite Sturm-Liouville problems. In this case, the weight function takes on both positive and negative signs on a given interval [a, b]. One feature of the non-definite Sturm-Liouville problem is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist.This thesis consists of three papers (papers A-C) and an introduction to this area, which puts these papers into a more general frame.In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper.In paper B we show that the interlacing property which holds in the one turning point case does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (−1, 2). We also present some theoretical results which support the numerical results.In paper C we extend results found in the paper by Jussi Behrndt et.al, in an essential way, to a case in which the weight function vanishes identically in a subinterval of [a, b]. In particular, we present some surprising numerical results on a specific problem in which the weight function is allowed to vanish identically on a subinterval of [−1, 2]. We also give some theoretical results which support these numerical examples.Godkänd; 2014; 20141110 (merkok); Nedanstående person kommer att hålla licentiatseminarium för avläggande av teknologie licentiatexamen. Namn: Mervis Kikonko Ämne: Matematik/Mathematics Uppsats: Qualitative and Spectral Theory of Some Regular Non-Definite Sturm-Liouville Problems Examinator: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Diskutant: Professor Sten Kaijser, Matematiska Institutionen, Uppsala Universitet Tid: Fredag den 11 december 2014 kl 10,00 Plats: E243, Luleå tekniska universite

Complex oscillations of non-definite Sturm-Liouville problems

We expand upon the basic oscillation theory for general boundary problems of the form $$-y''+q(t)y=\lambda r(t)y, \quad t \in I = [a,b]$$ where q and r are real-valued piecewise continuous functions and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. The non-definite case is characterized by the indefiniteness of each of the quadratic forms $$B+\int_a^b (|y'|^2 +q|y|^2)\quad \text{and}\quad \int_a^b r|y|^2,$$ over a suitable space where B is a boundary term. In 1918 Richardson proved that, in the case of the Dirichlet problem, if r(t) changes its sign exactly once and the boundary problem is non-definite then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2

Estimates on the lower bound of the eigenvalue of the smallest modulus associated with a general weighted Sturm-Liouville problem

We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by  Mingarelli (1988).Validerad; 2016; Nivå 2; 2016-10-19 (andbra)</p

Estimating the basic reproduction number for the 2015 bubonic plague outbreak in Nyimba district of Eastern Zambia.

BackgroundPlague is a re-emerging flea-borne infectious disease of global importance and in recent years, Zambia has periodically experienced increased incidence of outbreaks of this disease. However, there are currently no studies in the country that provide a quantitative assessment of the ability of the disease to spread during these outbreaks. This limits our understanding of the epidemiology of the disease especially for planning and implementing quantifiable and cost-effective control measures. To fill this gap, the basic reproduction number, R0, for bubonic plague was estimated in this study, using data from the 2015 Nyimba district outbreak, in the Eastern province of Zambia. R0 is the average number of secondary infections arising from a single infectious individual during their infectious period in an entirely susceptible population.Methodology/principal findingsSecondary epidemic data for the most recent 2015 Nyimba district bubonic plague outbreak in Zambia was analyzed. R0 was estimated as a function of the average epidemic doubling time based on the initial exponential growth rate of the outbreak and the average infectious period for bubonic plague. R0 was estimated to range between 1.5599 [95% CI: 1.382-1.7378] and 1.9332 [95% CI: 1.6366-2.2297], with average of 1.7465 [95% CI: 1.5093-1.9838]. Further, an SIR deterministic mathematical model was derived for this infection and this estimated R0 to be between 1.4 to 1.5, which was within the range estimated above.Conclusions/significanceThis estimated R0 for bubonic plague is an indication that each bubonic plague case can typically give rise to almost two new cases during these outbreaks. This R0 estimate can now be used to quantitatively analyze and plan measurable interventions against future plague outbreaks in Zambia