L\'evy mixing related to distributed order calculus, subordinators and slow diffusions

The study of distributed order calculus usually concerns about fractional derivatives of the form $\int_0^1 \partial^\alpha u \, m(d\alpha)$ for some measure $m$, eventually a probability measure. In this paper an approach based on L\'evy mixing is proposed. Non-decreasing L\'evy processes associated to L\'evy triplets of the form \l a(y), b(y), \nu(ds, y) \r are considered and the parameter $y$ is randomized by means of a probability measure. The related subordinators are studied from different point of views. Some distributional properties are obtained and the interplay with inverse local times of Markov processes is explored. Distributed order integro-differential operators are introduced and adopted in order to write explicitly the governing equations of such processes. An application to slow diffusions is discussed.Comment: in Journal of Mathematical Analysis and Applications (2015

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization with space-time grid for a parabolic diffusion problem and from the approximation of distributed order fractional equations. For this purpose we will use the classical GLT theory and the new concept of GLT momentary symbols. The first permits to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices, the latter permits to derive a function, which describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix-sizes but under given assumptions. The note is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques

Numerical analysis for distributed order differential equations

In this paper we present and analyse a numerical method for the solution of a distributed order differential equation

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques

International Conference on Dynamic Control and Optimization - DCO 2021: book of abstracts

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Controlo ótimo fracionário e aplicações biológicas

In this PhD thesis, we derive a Pontryagin Maximum Principle (PMP) for fractional optimal control problems and analyze a fractional mathematical model of COVID– 19 transmission dynamics. Fractional optimal control problems consist on optimizing a performance index functional subject to a fractional control system. One of the most important results in optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. First, we study properties of optimality for a dynamical system described by distributed-order non-local derivatives associated to a Lagrangian cost functional. We start by proving continuity and differentiability of solutions due to control perturbations. For smooth and unconstrained data, we obtain a weak version of Pontryagin's Maximum principle and a sufficient optimality condition under appropriate convexity. However, for controls taking values on a closed set, we use needle like variations to prove a strong version of Pontryagin's maximum principle. In the second part of the thesis, optimal control problems for fractional operators involving general analytic kernels are studied. We prove an integration by parts formula and a Gronwall inequality for fractional derivatives with a general analytic kernel. Based on these results, we show continuity and differentiability of solutions due to control perturbations leading to a weak version of the maximum principle. In addition, a wide class of combined fractional operators with general analytic kernels is considered. For this later problem, the control set is a closed convex subset of L2. Thus, using techniques from variational analysis, optimality conditions of Pontryagin type are obtained. Lastly, a fractional model for the COVID--19 pandemic, describing the realities of Portugal, Spain and Galicia, is studied. We show that the model is mathematically and biologically well posed. Then, we obtain a result on the global stability of the disease free equilibrium point. At the end we perform numerical simulations in order to illustrate the stability and convergence to the equilibrium point. For the data of Wuhan, Galicia, Spain, and Portugal, the order of the Caputo fractional derivative in consideration takes different values, characteristic of each region, which are not close to one, showing the relevance of the considered fractional models. 2020 Mathematics Subject Classification: 26A33, 49K15, 34A08, 34D23, 92D30.Nesta tese, derivamos o Princípio do Máximo de Pontryagin (PMP) para problemas de controlo ótimo fracionário e analisamos um modelo matemático fracionário para a dinâmica de transmissão da COVID-19. Os problemas de controlo ótimo fracionário consistem em otimizar uma funcional de índice de desempenho sujeita a um sistema de controlo fracionário. Um dos resultados mais importantes no controlo ótimo é o Princípio do Máximo de Pontryagin, que fornece uma condição de otimalidade necessária que toda a solução para o problema de otimização deve verificar. Primeiramente, estudamos propriedades de otimalidade para sistemas dinâmicos descritos por derivadas não-locais de ordem distribuída associadas a uma funcional de custo Lagrangiana. Começamos demonstrando a continuidade e a diferenciabilidade das soluções usando perturbações do controlo. Para dados suaves e sem restrições, obtemos uma versão fraca do princípio do Máximo de Pontryagin e uma condição de otimalidade suficiente sob convexidade apropriada. No entanto, para controlos que tomam valores num conjunto fechado, usamos variações do tipo agulha para provar uma versão forte do princípio do máximo de Pontryagin. Na segunda parte da tese, estudamos problemas de controlo ótimo para operadores fracionários envolvendo um núcleo analítico geral. Demonstramos uma fórmula de integração por partes e uma desigualdade Gronwall para derivadas fracionárias com um núcleo analítico geral. Com base nesses resultados, mostramos a continuidade e a diferenciabilidade das soluções por perturbações do controlo, levando a uma formulação de uma versão fraca do princípio do máximo de Pontryagin. Além disso, consideramos uma classe ampla de operadores fracionários combinados com núcleo analítico geral. Para este último problema, o conjunto de controlos é um subconjunto convexo fechado de L2. Assim, usando técnicas da análise variacional, obtemos condições de otimalidade do tipo de Pontryagin. Finalmente, estudamos um modelo fracionário da pandemia de COVID-19, descrevendo as realidades de Portugal, Espanha e Galiza. Mostramos que o modelo proposto é matematicamente e biologicamente bem colocado. Então, obtemos um resultado sobre a estabilidade global do ponto de equilíbrio livre de doença. No final, realizamos simulações numéricas para ilustrar a estabilidade e convergência do ponto de equilíbrio. Para os dados de Wuhan, Galiza, Espanha e Portugal, a ordem da derivada fracionária de Caputo em consideração toma valores diferentes característicos de cada região, e não próximos de um, mostrando a relevância de se considerarem modelos fracionários.Programa Doutoral em Matemática Aplicad

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Mathematical Economics

This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus

Stability Analysis of the Nabla Distributed-Order Nonlinear Systems

The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for the asymptotic stability of the nabla discrete distributed-order nonlinear systems is proposed based on Lyapunov direct method. In addition, some properties of the nabla distributed-order operators are derived. Based on these properties, a simpler criterion is provided to determine the stability of such systems. Finally, two examples are given to illustrate the validity of these results

Variable Order and Distributed Order Fractional Operators

Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined