Splinter Group:
Differential Equations and Numerical Analysis

Organizer: Niall Madden (NUI Galway)

All talks will be held in Room AC204.

Speakers include:

Tuesday, April 7 2009

14:15 - 14:45

Gabor Kiss (University of Bristol):
Delay-distribution effect on stability

We compare the stability regions of certain retarded functional differential equations. Namely, we consider the issue of what can be learnt from the stability of equations given with one delay about the stability of some associated equations given with distributed delays.

14:45 - 15:15

Aleksey Kostenko (Dublin Institute of Technology):
Spectral analysis of Sturm--Liouville operators with local point interactions

Spectral properties of operators H_{X,\alpha} and H_{X,\beta} associated in L^2(\Bbb R _+), respectively, with formal differential expressions l_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2}+\sum_{n\in \mathbb{N}}\alpha_n\delta(x-x_n),\qquad l_{X,\beta}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2}+\sum_{n\in \mathbb{N}}\beta_n\delta'(x-x_n),\quad \alpha_n,\ \beta_n\in\Bbb R, and subject to Dirichlet (or Neumann) boundary condition at x=0 are well studied in the case when the interactions sites are uniformly distributed, d_*:=\inf_{n\neq k}|x_n-x_k|>0 (numerous results as well as a comprehensive list of references may be found in a monograph of Albeverio, Gesztesy, Hoegh-Krohn, and Holden). If we drop the assumption d_*>0, then the analysis of operators H_{X,\alpha} and H_{X,\beta} becomes much more complicated. For instance, for the operator H_{X,\alpha} it is only known that it may be symmetric with nontrivial deficiency indices (the example of Christ and Stolz).
  The main aim of our talk is the spectral analysis of operators H_{X,\alpha} and H_{X,\beta} when the set X=\{x_n\}_{n=1}^\infty\subset\mathbb{R}_+ satisfies the assumptions d_*=0 and x_n\uparrow +\infty. We show that spectral properties of the operators H_{X,\alpha} and H_{X,\beta} are closely connected with the spectral properties of certain classes of unbounded Jacobi matrices. We exploit this connection to investigate self-adjointness, lower semi-boundedness, and discreteness of operators with local point interactions.

15:15 - 15:45

Georgi Grahovski (Dublin City University):
Fordy-Kulish Type Models and Spinor Bose-Einstein Condensates

Multicomponent generalizations of nonlinear Schrodinger-type models related to BD.I symmetric spaces are studied. Such models (known as Fordy-Kulish models) describe spinor Bose-Einstein condensates.
  These models are integrable by means of the inverse scattering method. Our analysis includes: the Hamiltonian properties of the models, an algebraic procedure to derive different types of soliton solutions, based on the Zakharov-Shabat dressing method.

16:30 - 17:00

Niall Madden (NUI Galway):
Numerical solution of a system of singularly perturbed reaction-diffusion equations by an overlapping Schwarz technique

We consider the analysis and application of an overlapping Schwarz algorithm to a coupled system of linear reaction-diffusion equations. We are particularly interested in how the algorithm should be extended to the case where solutions exhibit interacting boundary layers of different magnitudes. Emphasis is placed on establishing an upper bound for the number of iterations required.

17:00 - 17:30

Hani Benhassine (Institut Mathématique de Toulouse):
A FETI-like Domain Decomposition Domain Method For The Stokes Problem

The propose of this study is to apply a non overlapping domain decomposition method for solving the discrete problem resulting from a discretization of the incompressible Stokes equation by a mixed finite element method. In this way, the domain decomposition method can be seen as an efficient iterative procedure for solving the discrete problem and it applies to both a continuous or a discontinuous finite element approximation of the pressure. Convergence and stability proofs yield a theoretical background to the resulting numerical process when using a GMRES algorithm.