Browsing by Author "Torres, Delfim F. M."
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- Caputo derivatives of fractional variable order: numerical approximationsPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.
- Combined Fractional Variational Problems of Variable Order and Some Computational AspectsPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun.
- Constrained fractional variational problems of variable orderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.
- Fractional Herglotz variational problems of variable orderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.
- Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in PortugalPublication . Silva, Cristiana J.; Cruz, Carla; Torres, Delfim F. M.; Muñuzuri, Alberto P.; Carballosa, Alejandro; Area, Iván; Nieto, Juan J.; Fonseca-Pinto, Rui; Passadouro, Rui; Santos, Estevão Soares dos; Abreu, Wilson; Mira, JorgeThe COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a mode to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to “normal life” and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.
- Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable OrderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.