A rational spectral collocation method with adaptively transformed Chebyshev grid points

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Padé approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plan close to [-1,1]

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Optimal control of wave energy converters

Wave Energy Converters (WECs) are devices designed to absorb energy from ocean waves. The particular type of Wave Energy Converter (WEC) considered in this thesis is an oscillating body; energy conversion is carried out by means of a structure immersed in water which oscillates under forces exerted by waves. This thesis addresses the control of oscillating body WECs and the objective of the control system is to optimise the motion of the devices that maximises the energy absorption. In particular, this thesis presents the formulation of the optimal control problem for WECs in the framework of direct transcription methods, known as spectral and pseudospectral optimal control. Direct transcription methods transform continuous time optimal control problems into Non Linear Programming (NLP) problems, for which the literature (and the market) offer a large number of standard algorithms (and software packages). It is shown, in this thesis, that direct transcription gives the possibility of formulating complex control problems where realistic scenarios can be taken into account, such as physical limitations and nonlinearities in the behaviour of the devices. Additionally, by means of spectral and pseudospectral methods, it is possible to find an approximation of the optimal solution directly from sampled frequency and impulse response models of the radiation forces, obviating the need for finite order approximate models. By implementing a spectral method, convexity of the NLP problem, associated with the optimal control problem for a single body WEC described by a linear model, is demonstrated analytically. The solution to a nonlinear optimal control problem is approximated by means of pseudospectral optimal control. In the nonlinear case, simulation results show a significant difference in the optimal behaviour of the device, both in the motion and in the energy absorption, when the quadratic term describing the viscous forces are dominant, compared to the linear case. This thesis also considers the comparison of two control strategies for arrays of WECs. A Global Control strategy computes the optimal motion by taking into account the complete model of the array and it provides the global optimum for the absorbed energy. In contrast, an Independent Control strategy implements a control system on each device which is independent from all the other devices. The final part of the thesis illustrates an approach for the study of the effects of constraints on the total absorbed energy. The procedure allows the feasibility of the constrained energy maximisation problem to be studied, and it provides an intuitive framework for the design of WECs relating to the power take-off operating envelope, thanks to the geometrical interpretation of the functions describing both the total absorbed energy and the constraints

Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation

Tese de doutoramento. Ciências da Engenharia. 2006. Faculdade de Engenharia. Universidade do Porto, Instituto Superior Técnico. Universidade Técnica de Lisbo

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools