Fredholm determinants for the stability of travelling waves

This thesis investigates both theoretically and numerically the stability of travelling wave solutions using Fredholm determinants, on the real line. We identify a class of travelling wave problems for which the corresponding integral operators are of trace class. Based on the geometrical interpretation of the Evans function, we give an alternative proof connecting it to (modified) Fredholm determinants. We then extend that connection to the case of front waves by constructing an appropriate integral operator. In the context of numerical evaluation of Fredholm determinants, we prove the uniform convergence associated with the modified/regularised Fredholm determinants which generalises Bornemann's result on this topic. Unlike in Bornemann's result, we do not assume continuity but only integrability with respect to the second argument of the kernel functions. In support to our theory, we present some numerical results. We show how to compute higher order determinants numerically, in particular for integral operators belonging to classes I3 and I4 of the Schatten-von Neumann set. Finally, we numerically compute Fredholm determinants for some travelling wave problems e.g. the `good' Boussinesq equation and the fth-order KdV equation.UK EPSRC (Engineering and Physical Sciences Research Council) grant EP/G03613

Numerical analysis of some integral equations with singularities

In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.Supported by a college bursary from the University of Chester

Inverse problems and medical imaging: lecture notes

Lecture notes for the curricular unit of Inverse Problems and Medical Imaging (23036 - https://guiadoscursos.uab.pt/en/ucs/problemas-inversos-e-imagiologia-medica/ ) of the Doctor’s Degree in Applied Mathematics and Modelling of Universidade Aberta.info:eu-repo/semantics/draf

Numerical analysis of some integral equations with singularities

In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.EThOS - Electronic Theses Online ServiceUniversity of ChesterGBUnited Kingdo

Motion of a vortex sheet on a sphere with pole vortices

We cons i der the motion of a vortex sheet on the surface of a unit sphere in the presence of point vortices xed on north and south poles.Analytic and numerical research revealed that a vortex sheet in two-dimensional space has the following three properties.First,the vortex sheet is linearly unstable due to Kelvin-Helmholtz instability.Second,the curvature of the vortex sheet diverges in nite time.Last,the vortex sheet evolves into a rolling-up doubly branched spiral,when the equation of motion is regularized by the vortex method.The purpose of this article is to investigate how the curvature of the sphere and the presence of the pole vortices