SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

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

Novel Numerical Approaches for the Resolution of Direct and Inverse Heat Transfer Problems

This dissertation describes an innovative and robust global time approach which has been developed for the resolution of direct and inverse problems, specifically in the disciplines of radiation and conduction heat transfer. Direct problems are generally well-posed and readily lend themselves to standard and well-defined mathematical solution techniques. Inverse problems differ in the fact that they tend to be ill-posed in the sense of Hadamard, i.e., small perturbations in the input data can produce large variations and instabilities in the output. The stability problem is exacerbated by the use of discrete experimental data which may be subject to substantial measurement error. This tendency towards ill-posedness is the main difficulty in developing a suitable prediction algorithm for most inverse problems. Previous attempts to overcome the inherent instability have involved the utilization of smoothing techniques such as Tikhonov regularization and sequential function estimation (Beck’s future information method). As alternatives to the existing methodologies, two novel mathematical schemes are proposed. They are the Global Time Method (GTM) and the Function Decomposition Method (FDM). Both schemes are capable of rendering time and space in a global fashion thus resolving the temporal and spatial domains simultaneously. This process effectively treats time elliptically or as a fourth spatial dimension. AWeighted Residuals Method (WRM) is utilized in the mathematical formulation wherein the unknown function is approximated in terms of a finite series expansion. Regularization of the solution is achieved by retention of expansion terms as opposed to smoothing in the classical Tikhonov sense. In order to demonstrate the merit and flexibility of these approaches, the GTM and FDM have been applied to representative problems of direct and inverse heat transfer. Those chosen are a direct problem of radiative transport, a parameter estimation problem found in Differential Scanning Calorimetry (DSC) and an inverse heat conduction problem (IHCP). The IHCP is resolved for the cases of diagnostic deduction (discrete temperature data at the boundary) and thermal design (prescribed functional data at the boundary). Both methods are shown to provide excellent results for the conditions under which they were tested. Finally, a number of suggestions for future work are offered

Asymptotic behaviour of integro-differential equations describing clonal evolution of leukemia

This thesis is devoted to the analysis of a system of integro-differential equations describing leukemia, a type of blood cancer. Existence and uniqueness for arbitrary times are shown and the long-term behaviour of the solution is characterised. In order to achieve the asymptotic behaviour of the solution it is proved that a normalized (with respect to total mass) solution forms a Dirac sequence, thus the solution converges for time tending to infinity to a Dirac measure. Moreover the total mass converges, too, which is shown by combining an asymptotic stability result via a Lyapunov function with a perturbation argument. Additionally, the convergence result is generalised to a suitable measure space. Furthermore, the model is extended by an additional integral term with a small multiplicative coefficient in order to capture the idea of mutation. For this newly obtained system it is shown existence and uniqueness of both a solution for arbitrary times and a positive steady state. The latter is achieved by interpreting the steady state equations as an eigenvalue problem and by using the Krein-Rutman theorem. The local asymptotic stability of the steady state is proven by using linearised stability. The spectrum, which is crucial for linearised stability, is investigated with the method of the Weinstein-Aronszajn formula. Moreover, it is proven that the stable steady state of the extended model converges weakly^* to the stable steady state of the original model if the coefficient of the newly introduced integral term tends to zero. Lastly, a numerical scheme, which has been used to simulate the original model, is illustrated and its convergence to the analytical solution is proven

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

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