A signal processing view of differintegration

Proceedings of the European Control Conference, ECC’01, Porto, Portugal, September 2001The fractional differintegration problem is treated from the Signal Processing point of view. A brief review of the Laplace transform approach to differintegration is done. The continuoustime/ discrete-time system conversion is discussed and presented a Grünwald-Letnikov integration

Non-integer order derivatives

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2007Includes bibliographical references (leaves: 53-57)Text in English; Abstract: Turkish and Englishvii, 85 leavesThis thesis is devoted to integrals and derivatives of arbitrary order and applications of the described methods in various fields. This study intends to increase the accessibility of fractional calculus by combining an introduction to the mathematics with a review of selected recent applications in physics. It is described general definitions of fractional derivatives. This definitions are compared with their advantages and disadvantages. Fractional calculus concerns the generalization of differentiation and integration to non-integer (fractional) orders. The subject has a long mathematical history being discussed for the first time already in the correspondence of G. W. Leibnitz around 1690. Over the centuries many mathematicians have built up a large body of mathematical knowledge on fractional integrals and derivatives. Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. In the first chapter, Grünwald-Letnikov approache to generalization of the notion of the differentation and integration are considered. In the second chapter, the Riemann Liouville definition is given and it is compared with Grünwald-Letnikov definition. The last chapter, Caputo.s definition is given. In appendices, two applications are given including tomography and solution of Bessel equation

Finite difference methods for degenerate diffusion equations and fractional diffusion equations

This thesis focuses on the study of three mathematical models consisting of degenerate diffusion equations and fractional diffusion equations arising from different study needs. The first one is taken from the study of self-organized criticality phenomena arising from recent papers by Barbu. About the second one, that is connected to the obstacle problem, we analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. More precisely, we show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. At last, referring to the third one, that is a time-fractional type model (a Caputo time fractional degenerate diffusion equation) that can find a wider use, for example, from biology to mechanics, to superslow diffusion in porous media, till financial type phenomena, we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α ∈ (0, 1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. For all these models we provide numerical simulations

On the application of partial differential equations and fractional partial differential equations to images and their methods of solution

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

Stochastic quantisation of the fractional Φ34\Phi^4_3 model in the full subcritical regime

We construct the fractional $\Phi^4$ Euclidean quantum field theory on $R^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the non-linear stochastic partial differential equation describing the interacting field.Comment: 64 page

Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions

Perturbed Hamiltonians in one dimension: analysis for linear and nonlinear Schrödinger problems

The aim of this thesis is to present some results concerning with questions related to the perturbed linear and nonlinear Schrödinger problems. The perturbation is represented by a real potential that has to satisfy some integrability decay properties. These potentials are known in literature as "short range potentials". Moreover, the perturbed Hamiltonian has neither point spectrum nor zero resonance. The investigation focuses on the following main topics: To study the sectorial properties and the spectral scenario (modes and resonances) of the perturbed Hamiltonian (Chapter 1); to study how the classical homogeneous Besov and Sobolev spaces are transformed under the action of the wave operators (Chapter 2 - 3); to study the asymptotic behaviour of the solutions of the perturbed nonlinear Schrödinger problem with a scattering critical nonlinearity and small initial data (Chapter 4). Italian (Borsa finanziata dalla Regione Toscana nell’ambito del progetto “Pegaso”): Questa tesi contiene alcuni risultati connessi al problema di Schrödinger lineare e nonlineare quando l'Hamiltoniana classica viene perturbata. La perturbazione è rappresentata da un potenziale reale che soddisfa certe ipotesi di integrabilità e decadimento. Tali potenziali in letteratura sono noti come potenziali "a corto raggio d'azione". Inoltre, si suppone che l'Hamiltioniana perturbata non ammetta spettro puntuale né risonanze. Il lavoro di ricerca può essere essenzialmente sintatizzato nei seguenti punti: L'analisi delle proprietà settoriali e dello scenario spettrale per l'Hamiltoniana perturbata (Capitolo 1); lo studio della continuità degli operatori d'onda su spazi di Besov e Sobolev omogenei (Capitolo 2 - 3); l'analisi del comportamento asintotico delle soluzioni del problema di Schrödinger nonlineare perturbato con linearità critica e per dati iniziali piccoli in opportune norme di Sobolev (Capitolo 4)