MS-Th-BC-09
Applied and computational complex analysis I

Date: August 13
Time: 10:00--12:00
Room: 203A

(Note: Click title to show the abstract.)

Organizer:
Pelloni, Beatrice (Univ. of Reading)

Abstract: The focus of the session is on fundamental complex analysis methods for exploring the deep mathematical connections between shapes, singularities of complex functions, and spectra.
We have assembled a list of researchers that have in common the use of tools of complex analysis for uncovering the underlying structure of mathematical and physical phenomena. Such tools are fundamental, for example, in the study of boundary value problems and differential operators via the Unified Transform of Fokas, in the derivation of integral transforms for medical imaging, in Crowdy¡¯s work on the evolution of shapes in boundary value problems, such as in the famous Hele-Shaw problem, and
in the study of the geometric patterns arising in the study of singularities of
certain distinguished ODEs..
The full impact of such approaches is only beginning to be felt and in this symposium the speakers will discuss advances and open challenges, theoretical and numerical.
This is one minisymposium in a series of two, organised by the network ACCA-UK and ACCA-Japan.

MS-Th-BC-09-1
10:00--10:30
Fast solution of the boundary integral equations with the generalized Neumann kernel
Nasser, Mohamed M S (King Khalid Uinversity)
Abstract: We briefly review the derivation and the properties of the boundary integral equations with the generalized Neumann kernel which have been derived in the past decade by the author and his collaborators. We then present a fast numerical method for solving these integral equations. The presented method gives accurate results even for domains of very high connectivity, domains with piecewise smooth boundaries, domains with close-to-touching boundaries, and domains of real world problems.

MS-Th-BC-09-2
10:30--11:00
Augmented eigenfunctions: a new spectral object appearing in the integral representation of the solution of linear initial-boundary value problems
Smith, David (Univ. of Cincinnati)
Abstract: We study initial-boundary value problems for linear, equations of arbitrary order, on a finite or semi-infinite domain, with arbitrary boundary conditions, using the Unified Transform Method of Fokas. The solution obtained is expressed as an integral, which represents a new kind of spectral transform. We compare the new solution representation, with classical Fourier transform solution representations. In doing so, we discover a new species of spectral object.

MS-Th-BC-09-3
11:00--11:30
Computation of an infinite integral with unbounded integrand
Ooura, Takuya (Kyoto Univ)
Abstract: The infinite integral $\int_{0}^{\infty} xdx/(1+x^6 \sin^2 x)$ has an unbounded integrand but is convergent. Computing the value of this integral has been a problem since 1984. We demonstrate that the method using the Hilbert transform to cancel the singularity of the integrand and compute the value of the integral with high accuracy using a superconvergent double exponential quadrature.

MS-Th-BC-09-4
11:30--12:00
Fourier-Mellin transforms for multiply connected circular domains and applications
Crowdy, Darren (Imperial College London)
Abstract: In this talk we present new transform pairs for analytic functions defined in generally multiply connected circular domains. Circular domains are those with boundaries made up of circular arcs or straight lines. The transform pairs, which will be shown to be the natural generalizations of classical Fourier and Mellin transforms, can be used to solve boundary value problems for harmonic and biharmonic fields that are not amenable to solution by traditional techniques.

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