Numerical approximation of the non-essential spectrum of abstract delay differential equations

Abstract Delay Differential Equations (ADDEs) extend Delay Differential Equations (DDEs) from finite to infinite dimension. They arise in many application fields. From a dynamical system point of view, the stability analysis of an equilibrium is the first relevant question, which can be reduced to the stability of the zero solution of the corresponding linearized system. In the understanding of the linear case, the essential and the non-essential spectra of the infinitesimal generator are crucial. We propose to extend the infinitesimal generator approach developed for linear DDEs to approximate the non-essential spectrum of linear ADDEs. We complete the paper with the numerical results for a homogeneous neural field model with transmission delay of a single population of neurons

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Complete Controllability of Fractional Neutral Differential Systems in Abstract Space

By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us

Quantum Biomimetics

136 p.En esta tesis proponemos el concepto de Biomimética Cuántica orientado hacia la reproducción de comportamientos propios de los seres vivos en protocolos de información cuántica. En concreto, las propiedades que aspiramos a imitar emergen como resultado de fenómenos de interacción en diferentes escalas, resultando inaccesibles para un tratamiento matemático acorde al ofrecido por las plataformas de tecnologías cuánticas. Por tanto, el objetivo de la tesis es el de diseñar modelos con cabida para las mencionadas características biológicas pero simplificados de forma que puedan ser adaptados en protocolos experimentales. La tesis se divide en tres partes, una por cada rasgo biológico diferente empleado como inspiración: selección natural, memoria e inteligencia. El estudio presentado en la primera parte culmina con la obtención de un modelo de vida artificial con una identidad exclusivamente cuántica, que no solo permite la escenificación del modelo de selección natural a escala microscópica si no que proporciona un posible marco para la implementación de algoritmos genéticos y problemas de optimización en plataformas cuánticas. En la segunda parte se muestran algoritmos asociados con la simulación de evolución temporal regida por ecuaciones con una dependencia explicita en términos deslocalizados temporalmente. Estos permiten la incorporación de la retroalimentación y posalimentación al conjunto de herramientas en información cuántica. La tercera y última parte versa acerca de la posible simbiosis entre los algoritmos de aprendizaje y los protocolos cuánticos. Mostramos como aplicar técnicas de optimización clásicas para tratar problemas cuánticos así como la codificación y resolución de problemas en dinámicas puramente cuánticas

On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations. Most of these equations are highly nonlinear and exact solutions are not always possible. Exact solutions always give a good account of the physical nature of the phenomena modeled. However, existing analytical methods can only handle a limited range of these equations. Semi-numerical and numerical methods give approximate solutions where exact solutions are impossible to find. However, some common numerical methods give low accuracy and may lack stability. In general, the character and qualitative behaviour of the solutions may not always be fully revealed by numerical approximations, hence the need for improved semi-numerical methods that are accurate, computational efficient and robust. In this study we introduce innovative techniques for finding solutions of highly nonlinear coupled boundary value problems. These techniques aim to combine the strengths of both analytical and numerical methods to produce efficient hybrid algorithms. In this work, the homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral methods are well known for their high levels of accuracy. The new spectral homotopy analysis method is further improved by using a more accurate initial approximation to accelerate convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral methods are used to solve the linearised equations. The new techniques were used to solve mathematical models in fluid dynamics. The thesis comprises of an introductory Chapter that gives an overview of common numerical methods currently in use. In Chapter 2 we give an overview of the methods used in this work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional squeezing flow of a viscous fluid between two approaching parallel plates and the steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter 4 the methods were used to find solutions of the laminar heat transfer problem in a rotating disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem due to a shrinking sheet with a chemical reaction, were solved using the new methods

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

On functional differential equations associated to controlled structures with propagation

The method of integration along the characteristics has turned to be quite fruitful for qualitative analysis of physical and engineering systems described by large classes of partial differential equations of hyperbolic type in the plane (time and one space dimension) with real characteristics. In this paper there is presented an overview of such models under the aforementioned approach. We mention in this abstract the models describing transport phenomena (e.g. circulating fuel nuclear reactors and tubular reactors of the biotechnology) and propagation phenomena (e.g. electrical transmission lines such as waveguides or water, steam and gas pipes). In the first case (transport phenomena) there exists a single forward (progressive) wave due to the fact that there exists a single family of characteristics which are increasing. In the second case (propagation) there are to be met both forward (progressive) and backward (reflected) waves and two families of characteristics. In the nonlinear case the systems of conservation laws belong to both categories of systems. Integration along the characteristics allows association of some systems of functional (differential) equations; a one-to-one (injective) correspondence between the solutions of the two mathematical objects (the boundary value problem for the partial differential equations and the functional equations) is established such that all properties obtained for one of them is projected back on the other. In this way continuous and discontinuous classical solutions can be analyzed from the point of view of the well-posedness in the sense of Hadamard (existence, uniqueness and data/parameter dependence), existence of some invariant sets and stability. The various functional equations thus introduced are mathematical objects interesting for themselves such as the neutral functional differential equations which appear in lossless and distortionless wave propagation when differential equations are to be met in the boundary conditions

Differentiable positive definite kernels on two-point homogeneous spaces

In this work we study continuous kernels on compact two-point homogeneous spaces which are positive definite and zonal (isotropic). Such kernels were characterized by R. Gangolli some forty years ago and are very useful for solving scattered data interpolation problems on the spaces. In the case the space is the d-dimensional unit sphere, J. Ziegel showed in 2013 that the radial part of a continuous positive definite and zonal kernel is continuously differentiable up to order ⌊(d−1)/2⌋ in the interior of its domain. The main issue here is to obtain a similar result for all the other compact two-point homogeneous spaces.CNPq (grant 141908/2015-7)FAPESP (grant 2014/00277-5