Books
56.
The variable order fractional calculus of variations
Almeida, Ricardo and Tavares, Dina and Torres, Delfim F. M.
Springer
This book intends to deepen the study of the fractional calculus, giving special
emphasis to variableorder operators
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Book Chapters
55.
Noninvasive control of the fractional HegselmannKrause type model
Almeida, Ricardo and Malinowska, A.B. and Odzijewicz, T.
NonInteger Order Calculus and its Applications. Lecture Notes in Electrical Engineering
Springer
In this paper, the fractional order Hegselmann–Krause type model with leadership is studied. We seek an optimal control strategy for the system to reach a consensus in such a way that the control mechanism is included in the leader dynamics. Necessary optimality conditions are obtained by the use of a fractional counterpart of Pontryagin Maximum Principle. The effectiveness of the proposed control strategy is illustrated by numerical examples.
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54.
A survey on fractional variational calculus
Almeida, Ricardo and Torres, Delfim F. M.
Handbook of Fractional Calculus with Applications
De Gruyter
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of EulerLagrange type for the fundamental, higherorder, and isoperimetric problems, and we compute approximate solutions based on truncated GrünwaldLetnikov approximations of the Caputo derivatives.
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Peer Reviewed
53.
Analysis of fractional integrodifferential equations of thermistor type
Sidi Ammi, Moulay Rchid and Torres, Delfim F. M.
Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory
De Gruyter
We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of
fractional order. As an illustrative example, we review results as regards fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: problems concerned with Riemann–Liouville, Caputo, and timescale fractional operators. The existence and uniqueness of positive solutions are obtained through suitable fixedpoint theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.
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Peer Reviewed
52.
Variable order Mittag–Leffler fractional operators on isolated time scales and application to the calculus of variations
Abdeljawad, Thabet and Mert, Raziye and Torres, Delfim F. M.
Fractional Derivatives with MittagLeffler Kernel. Studies in Systems, Decision and Control
Springer
We introduce new fractional operators of variable order in isolated time scales with Mittag–Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variableorder difference operators. Main
results give fractional integration by parts formulas and necessary optimality conditions of Euler–Lagrange type.
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Peer Reviewed
51.
Timefractional optimal control of initial value problems on time scales
Bahaa, Gaber M. and Torres, Delfim F. M.
Nonlinear Analysis and Boundary Value Problems. NABVP 2018. Springer Proceedings in Mathematics & Statistics
Springer
We investigate Optimal Control Problems (OCP) for fractional systems involving fractionaltime derivatives on time scales. The fractionaltime derivatives and integrals are considered, on time scales, in the Riemann–Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a deltaintegral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler–Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann–Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.
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Peer Reviewed
50.
O Aeroporto Francisco Sá Carneiro e a sua relação com o turismo na Região Norte de Portugal
Barros, Helena de and Alonso, Hugo
26th APDR Congress
Associação Portuguesa para o Desenvolvimento Regional
O Aeroporto Francisco Sá Carneiro e o turismo na região Norte de Portugal têm crescido de forma muito significativa nos últimos anos. Este artigo apresenta um estudo da relação entre o número de passageiros que circulam no aeroporto e as dormidas na região. O estudo é baseado numa análise de regressão linear simples. Como resultado, propõese uma forma de prever as dormidas, conhecido o número de passageiros.
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49.
New inequalities for ηquasiconvex function
Nwaeze, Eze R. and Torres, Delfim F. M.
Frontiers in Functional Equations and Analytic Inequalities
Springer
The class of ηquasiconvex functions was introduced in 2016. Here we
establish novel inequalities of Ostrowski type for functions whose second derivative,
in absolute value raised to the power q ≥ 1, is ηquasiconvex. Several interesting
inequalities are deduced as special cases. Furthermore, we apply our results to the
arithmetic, geometric, Harmonic, logarithmic, generalized log and identric means,
getting new relations amongst them.
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Peer Reviewed
48.
Harmonic and trace inequalities in Lipschitz domains
Touhami, Soumia and Chaira, Abdellatif and Torres, Delfim F. M.
Frontiers in Functional Equations and Analytic Inequalities
Springer
We prove boundary inequalities in arbitrary bounded Lipschitz domains on the trace space of Sobolev spaces. For that, we make use of the trace operator, its Moore–Penrose inverse, and of a special inner product. We show that our trace inequalities are particularly useful to prove harmonic inequalities, which serve as powerful tools to characterize the harmonic functions on Sobolev spaces of noninteger order.
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Peer Reviewed
Articles
47.
Periodic statespace representations of periodic convolutional codes
Napp, Diego and Pereira, Ricardo and Pinto, Raquel and Rocha, Paula
Cryptography and Communications
Springer Verlag
In this paper we study the representation of periodically timevarying convolutional codes by means of periodic inputstateoutput models. In particular, we focus on period two and investigate under which conditions a given twoperiodic convolutional code (obtained by alternating two timeinvariant encoders) can be represented by a periodic inputstateoutput system. We first show that one cannot expect, in general, to obtain a periodic inputstateoutput representation of a periodic convolutional code by means of the individual realizations of each of the associated timeinvariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic statespace realization that is minimal. Finally, examples to illustrate the results are presented.
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Peer Reviewed
46.
Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems
Salati, Abubakar Bello and Shamsi, Mostafa and Torres, Delfim F. M.
Communications in Nonlinear Science and Numerical Simulation
Elsevier
This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many welldeveloped algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bangbang FOCP and an optimal control of a fractionalorder HIVimmune system.
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Peer Reviewed
45.
Solutions of systems with the CaputoFabrizio fractional delta derivative on time scales
Mozyrska, Dorota and Torres, Delfim F. M. and Wyrwas, Malgorzata
Nonlinear Analysis: Hybrid Systems
Elsevier
CaputoFabrizio fractional delta derivatives on an arbitrary time scale are
presented. When the time scale is chosen to be the set of real numbers, then
the CaputoFabrizio fractional derivative is recovered. For isolated or partly
continuous and partly discrete, i.e., hybrid time scales, one gets new
fractional operators. We concentrate on the behavior of solutions to initial
value problems with the CaputoFabrizio fractional delta derivative on an
arbitrary time scale. In particular, the exponential stability of linear
systems is studied. A necessary and sufficient condition for the exponential
stability of linear systems with the CaputoFabrizio fractional delta
derivative on time scales is presented. By considering a suitable fractional
dynamic equation and the Laplace transform on time scales, we also propose a
proper definition of CaputoFabrizio fractional integral on time scales.
Finally, by using the Banach fixed point theorem, we prove existence and
uniqueness of solution to a nonlinear initial value problem with the
CaputoFabrizio fractional delta derivative on time scales.
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44.
The spread of a financial virus through Europe and beyond
Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.
AIMS Mathematics
AIMS Press
We analyse the importance of international relations between countries on the financial stability. The contagion effect in the network is tested by implementing an epidemiological model, comprising a number of European countries and using bilateral data on foreign claims between them. Banking statistics of consolidated foreign claims on ultimate risk bases, obtained from the Banks of International Settlements, allow us to measure the exposure of contagion spreading from a particular country to the other national banking systems. We show that the financial system of some countries, experiencing the debt crisis, is a source of global systemic risk because they threaten the stability of a larger system, being a global threat to the intoxication of the world economy and resulting in what we call a `financial virus'. Illustrative simulations were done in the NetLogo multiagent programmable modelling environment and in MATLAB.
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Peer Reviewed
43.
Column distances of convolutional codes over Z_p^r
Napp, Diego and Pinto, Raquel and Toste, Marisa
IEEE Transactions on Information Theory
Institute of Electrical and Electronics Engineers
Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of pencoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.
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42.
Existence of solution to a nonlocal conformable fractional thermistor problem
Moulay Rchid Sidi Ammi and Torres, Delfim F. M.
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Faculty of Sciences University of Ankara
We study a nonlocal thermistor problem for fractional derivatives in the conformable sense. Classical Schauder's fixed point theorem is used to derive
the existence of a tube solution.
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Peer Reviewed
41.
A spacetime pseudospectral discretization method for solving diffusion optimal control problems with twosided fractional derivatives
Mushtaq Salh Ali and Mostafa Shamsi and Hassan KhosravianArab and Torres, Delfim F. M. and Farid Bozorgnia
Journal of Vibration and Control
SAGE Publications
We propose a direct numerical method for the solution of an optimal control problem governed by a twoside spacefractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre–Gauss–Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low central processing unit time.
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Peer Reviewed
40.
Fractional order version of the HJB equation
Razminia, Abolhassan and AsadiZadehShiraz, Mehdi and Torres, Delfim F. M.
Journal of Computational and Nonlinear Dynamics
American Society of Mechanical Engineers
We consider an extension of the wellknown HamiltonJacobiBellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.
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39.
Structural derivatives on time scales
Bayour, Benaoumeur and Torres, Delfim F. M.
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Faculty of Sciences University of Ankara
We introduce the notion of structural derivative on time scales. The new
operator of differentiation unifies the concepts of fractal and fractional
order derivative and is motivated by lack of classical differentiability of
some selfsimilar functions. Some properties of the new operator are proved and
illustrated with examples.
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38.
Functional characterizations of trace spaces in Lipschitz domains
Touhami, Soumia and Chaira, Abdellatif and Torres, Delfim F. M.
Banach Journal of Mathematical Analysis
Duke University Press
Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces Hs (∂Ω) involving a family of positive selfadjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary ∂Ω of a bounded Lipschitz domain Ω ⊂ R d
and applying Moore–Penrose pseudoinverse properties together with a special inner product on H1 (Ω). We also establish generalized results of the Moore– Penrose pseudoinverse.
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Peer Reviewed
37.
Exact solution to a dynamic SIR model
Bohner, Martin and Streipert, Sabrina and Torres, Delfim F. M.
Nonlinear Analysis: Hybrid Systems
Elsevier
We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to timedependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible–infected–removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.
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Peer Reviewed
36.
A sufficient optimality condition for delayed statelinear optimal control problems
LemosPaião, Ana Pedro and Silva, Cristiana J. and Torres, Delfim F. M.
Discrete and Continuous Dynamical Systems  Series B
American Institute of Mathematical Sciences
We give answer to an open question by proving a sufficient optimality condition for statelinear optimal control problems with time delays in state
and control variables. In the proof of our main result, we transform a delayed statelinear optimal control problem to an equivalent nondelayed problem. This allows us to use a wellknown theorem that ensures a sufficient optimality condition for nondelayed statelinear optimal control problems. An example is given in order to illustrate the obtained result.
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Peer Reviewed
35.
On duals and paritychecks of convolutional codes over Z p r
El Oued, Mohamed and Napp, Diego and Pinto, Raquel and Toste, Marisa
Finite fields and their applications
Elsevier
A convolutional code C over Z_{p^r}((D)) is a Z_{p^r}((D))submodule of Z_{p^r}^n((D)) that admits a polynomial set of generators, where Z_{p^r}((D)) stands for the ring of (semiinfinity) Laurent series. In this paper we study several structural properties of its dual C^{perp} . We use these results to provide a constructive algorithm to build an explicit generator matrix of C^{perp}. Moreover, we show that the transpose of such a matrix is a paritycheck matrix (also called syndrome former) of C.
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Peer Reviewed
34.
Analysis and numerical approximation of tempered fractional calculus of variations problems
Almeida, Ricardo and Morgado, M. Luísa
Journal of Computational and Applied Mathematics
Elsevier
In this paper, we study variational problems where the cost functional involves the tempered Caputo fractional derivative. Several important optimization conditions are derived to find the optimal solution. Sufficient and necessary conditions are presented for different variational problems. For example, the cases of integral (isoperimetric problem) and holonomic constraints are considered, as well as problems with high order derivatives. A numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples
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Peer Reviewed
33.
The risk of contagion spreading and its optimal control in the economy
Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.
Statistics, Optimization and Information Computing
International Academic Press
The global crisis of 2008 provoked a heightened interest among scientists to study the phenomenon, its
propagation and negative consequences. The process of modelling the spread of a virus is commonly used in epidemiology.
Conceptually, the spread of a disease among a population is similar to the contagion process in economy. This similarity
allows considering the contagion in the world financial system using the same mathematical model of infection spread that
is often used in epidemiology. Our research focuses on the dynamic behaviour of contagion spreading in the global financial
network. The effect of infection by a systemic spread of risks in the network of national banking systems of countries is
tested. An optimal control problem is then formulated to simulate a control that may avoid significant financial losses. The
results show that the proposed approach describes well the reality of the world economy, and emphasizes the importance of
international relations between countries on the financial stability.
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32.
Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model
Duarte, Jorge and Martins, Nuno and Rogovchenko, Svitlana and Rogovchenko, Yuriy and Januário, Cristina
Journal of Mathematical Biology
Springer
Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramatically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical SusceptibleInfectedRecovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model.
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Peer Reviewed
31.
An epidemiological MSEIR model described by the Caputo fractional derivative
Almeida, Ricardo and Cruz, Artur Miguel and Martins, Natália and Monteiro, Maria Teresa
International Journal of Dynamics and Control
Springer
A fractional MSEIR model is presented, involving the Caputo fractional derivative. The equilibrium points and the basic reproduction number are computed. An analysis of the local asymptotic stability at the disease free equilibrium is given. Finally a numerical simulation, using Matlab based on optimization techniques, of the varicella outbreak among Shenzhen school children, China, is carried out.
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Peer Reviewed
30.
A numerical approach for solving fractional optimal control problems using modified hat functions
Nemati, Somayeh and Lima, Pedro M. and Torres, Delfim F. M.
Communications in Nonlinear Science and Numerical Simulation
Elsevier
We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and
the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the RiemannLiouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and RiemannLiouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of
fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations,
and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.
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Peer Reviewed
29.
A collocation method of lines for twosided spacefractional advectiondiffusion equations with variable coefficients
Almoaeet, Mohammed K. and Shamsi, Mostafa and KhosravianArab, Hassan and Torres, Delfim F. M.
Mathematical Methods in the Applied Sciences
Wiley
We present the method of lines (MOL), which is based on the spectral collocation method, to solve spacefractional advectiondiffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left and rightsided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method
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Peer Reviewed
28.
A numerical study of fractional relaxation–oscillation equations involving ψCaputo fractional derivative
Almeida, Ricardo and Jleli, Mohamed and Samet, Bessem
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
Springer
We provide a numerical method to solve a certain class of fractional differential equations involving ψ Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce ψ shifted Legendre polynomial basis, and we derive an explicit formula for the ψ fractional integral of ψ shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.
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Peer Reviewed
27.
Stability of a fractional HIV/AIDS model
Silva, Cristiana J. and Torres, Delfim F. M.
Mathematics and Computers in Simulation
Elsevier
We propose a fractional order model for HIV/AIDS transmission. Local and uniform stability of the fractional order model is studied. The theoretical results are illustrated through numerical simulations.
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Peer Reviewed
26.
A sufficient optimality condition for nonlinear delayed optimal control problems
LemosPaião, Ana P. and Silva, Cristiana J. and Torres, Delfim F. M.
Pure and Applied Functional Analysis
Yokohama Publishers
We prove a sufficient optimality condition for nonlinear optimal control
problems with delays in both state and control variables. Our result requires
the verification of a HamiltonJacobi partial differential equation and is
obtained through a transformation that allow us to rewrite a delayed optimal
control problem as an equivalent nondelayed one.
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25.
Fractional differential equations with mixed boundary conditions
Almeida, Ricardo Miguel
Bulletin of the Malaysian Mathematical Sciences Society
Springer
In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order α ∈ (2, 3), involving a general form of fractional derivative. First, we prove an equivalence
between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples.
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Peer Reviewed
24.
Some inequalities for intervalvalued functions on time scales
Zhao, Dafang and Ye, Guoju and Liu, Wei and Torres, Delfim F. M.
Soft Computing
Springer
We introduce the interval Darboux delta integral (shortly, the IDΔ integral) and the interval Riemann delta integral (shortly, the IR Δ integral) for intervalvalued functions on time scales. Fundamental properties of ID and IR Δ integrals and examples are given. Finally, we prove Jensen’s, Hölder’s and Minkowski’s inequalities for the IR Δ integral. Also, some examples are given to illustrate our theorems
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23.
Optimal leader–follower control for the fractional opinion formation model
Almeida, Ricardo and Malinowska, Agnieszka B. and Odzijewicz, Tatiana
Journal of Optimization Theory and Applications
Springer
This paper deals with an opinion formation model, that obeys a nonlinear system of fractionalorder differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader–follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented.
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22.
Caputo fractional differential equation with state dependent delay and practical stability
Agarwal, Ravi and Almeida, Ricardo and Hristova, Snezhana and O'Regan, Donal
Dynamic Systems and Applications
Dynamic Publishers, Inc
Practical stability properties of Caputo fractional delay differential equations is studied and, in particular, the case with state dependent delays is considered. These type of delays is a generalization of several types of delays such as constant delays, time variable delays, or distributed delays. In connection with the
presence of a delay in a fractional differential equation and the application of the fractional generalization of the Razumikhin method, we give a brief overview of the
most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. Three types of derivatives for Lyapunov functions, the Caputo fractional derivative, the Dini fractional derivative, and the Caputo fractional Dini derivative, are applied to obtain several sufficient conditions for practical stability. An appropriate Razumikhin condition is applied. These derivatives allow the application of nonquadratic Lyapunov function for studying stability properties. We illustrate our theory on several nonlinear Caputo fractional differential equations with different types of delays
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21.
Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives
Ammi, Moulay Rchid Sidi and Torres, Delfim F. M.
Computers and Mathematics with Applications
Elsevier
We study an optimal control problem associated to a fractional nonlocal
thermistor problem involving the ABC (AtanganaBaleanuCaputo) fractional time
derivative. We first prove the existence and uniqueness of solution. Then, we
show that an optimal control exists. Moreover, we obtain the optimality system
that characterizes the control.
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20.
A finite element approximation for a class of Caputo timefractional diffusion equations
Ammi, Moulay Rchid Sidi and Jamiai, Ismail and Torres, Delfim F. M.
Computers and Mathematics with Applications
Elsevier
We develop a fully discrete scheme for timefractional diffusion equations by
using a finite difference method in time and a finite element method in space.
The fractional derivatives are used in Caputo sense. Stability and error
estimates are derived. The accuracy and efficiency of the presented method is
shown by conducting two numerical examples.
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19.
Leaderfollowing consensus for fractional multiagent systems
Almeida, Ricardo and Girejko, Ewa and Hristova, Snezhana and Malinowska, Agnieszka B.
Advances in Difference Equations
SpringerOpen
A leaderfollowing consensus for Caputo fractional multiagent systems with
nonlinear intrinsic dynamics is investigated. The second Lyapunov method is used to
design a control protocol ensuring a consensus for two types of multiagent systems.
Contrary to the previous studies on leaderfollowing consensus, the investigation
covers systems with bounded and unbounded timedependent Lipschitz coefficients
in the intrinsic dynamics. Moreover, coupling strength describing the interactions
between agents is considered to be a function of time.
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18.
The Portuguese Meeting in Biomathematics
Torres, Delfim F. M. and Area, Iván and Silva, César and Silva, Cristiana J.
Statistics, Optimization and Information Computing
IAPress
The main contributions of [Stat. Optim. Inf. Comput. Vol. 7, No. 3 (2019)], consisting of 7 papers selected and
revised from the 2nd Portuguese Meeting in Biomathematics (EPB’2018), are highlighted.
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17.
Analysis of a SIRI epidemic model with distributed delay and relapse
Elazzouzi, Abdelhai and Alaoui, Abdesslem Lamrani and Tilioua, Mouhcine and Torres, Delfim F. M.
Statistics, Optimization and Information Computing
IAPress
We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R0 < 1. Moreover,we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R0 > 1.
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16.
A minimal HIVAIDS infection model with general incidence rate and application to Morocco data
Lotfi, El Mehdi and Mahrouf, Marouane and Maziane, Mehdi and Silva, Cristiana J. and Torres, Delfim F. M. and Yousfi, Noura
Statistics, Optimization and Information Computing
IAPress
We study the global dynamics of a SICA infection model with general incidence rate. The proposed model is calibrated with cumulative cases of infection by HIV–AIDS in Morocco from 1986 to 2015. We first prove that our model is biologically and mathematically wellposed. Stability analysis of different steady states is performed and threshold parameters are identified where the model exhibits clearance of infection or maintenance of a chronic infection. Furthermore, we examine the robustness of the model to some parameter values by examining the sensitivity of the basic reproduction number. Finally, using numerical simulations with real data from Morocco, we show that the model predicts well such reality.
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15.
Optimal control and sensitivity analysis of a fractional order TB model
Rosa, Silvério and Torres, Delfim F. M.
Statistics, Optimization and Information Computing
A Caputo fractionalorder mathematical model for the transmission dynamics of tuberculosis (TB) was recently proposed in [Math. Model. Nat. Phenom. 13 (2018), no. 1, Art. 9]. Here, a sensitivity analysis of that model is done, showing the importance of accuracy of parameter values. A fractional optimal control (FOC) problem is then formulated and solved,with the rate of treatment as the control variable. Finally, a costeffectiveness analysis is performed to assess the cost and the effectiveness of the control measures during the intervention, showing in which conditions FOC is useful with respect to classical (integerorder) optimal control.
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14.
Influence of the topology on the dynamics of a complex network of HIV/AIDS epidemic models
Cantin, Guillaume and Silva, Cristiana J.
AIMS Mathematics
AIMS Press
In this paper, we propose an original complex network model for an epidemic problem in an heterogeneous geographical area. The complex network is constructed by coupling nonidentical instances of a HIV/AIDS epidemiological model for which a diseasefree equilibrium and an endemic equilibrium can coexist. After proving the existence of a positively invariant region for the solutions of the complex network problem, we investigate the effect of the coupling on the dynamics of the network, and establish the existence of a unique diseasefree equilibrium for the whole network, which is globally asymptotically stable. We prove the existence of an optimal topology that minimizes the level of infected individuals, and apply the theoretical results to the case of the Cape Verde archipelago.
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13.
Realization of 2D (2,2)periodic encoders by means of 2D periodic separable Roesser models
Pereira, Ricardo and Napp, Diego and Pinto, Raquel and Rocha, Paula
International Journal of Applied Mathematics and Computer Science
University of Zielona Gora Press
It is wellknown that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. In comparison to the literature on onedimensional (1D) timeinvariant convolutional codes, there exists only relatively few results on the realization problem for the timevarying 1D convolutional codes and even fewer if the convolutional codes are twodimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimility and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.
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12.
Towards a geometric theory for nD behaviors: conditioned invariance and detectability subspaces
Pereira, Ricardo and Rocha, Paula
IFACPapersOnLine
Elsevier
We introduce the definitions of observer, conditioned invariance and detectability subspaces for discrete multidimensional behavioral systems, based on our previous work for the continuous 1D case, as a step forward in the attempt to develop a geometric theory for nD behaviors.
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11.
Fractional differential equations and Volterra–Stieltjes integral equations of the second kind
Asanov, Avyt and Almeida, Ricardo and Malinowska, Agnieszka B.
Computational and Applied Mathematics
Springer
In this paper, we construct a method to find approximate solutions to fractional differential
equations involving fractional derivatives with respect to another function. The method is
based on an equivalence relation between the fractional differential equation and the Volterra–
Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to
solve numerically the integral equation and an estimation for the error is given. Results of
numerical experiments demonstrate that satisfactory and reliable results could be obtained
by the proposed method.
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10.
Preface of the “4th Symposium on Modelling and Simulation in Computer Sciences and Engineering”
Miranda, Francisco and Abreu, Carlos and Miranda, Daniel
AIP Conference Proceedings
AIP Publishing
The 4th Symposium on Modelling and Simulation in Computer Sciences and Engineering was held in the 16th
International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2018), Rhodes, Greece, 1318
September 2018.
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9.
Theoretical simulation of the influence of cathode formulation on lithiumion battery performance
Miranda, D. and Miranda, F. and Almeida, A. M. and LancerosMéndez, S. and Costa, C. M.
AIP Conference Proceedings
AIP Publishing
Optimizing cathode electrode formulation is essential in the development and performance of lithiumion
batteries, as the cathode affects the capacity of the battery. Cathode electrode formulation is based on different materials
and relative contents. In this work, the cathode performance for the LiMn2O4, LMO, active material has been obtained by
theoretical simulations for different materials formulations and at various discharge rates. Further, the simulations were
compared with experimental results. It is demonstrated that the optimization of the electrode formulation strongly
depends on the percentage of conductive material, existing a minimum conductive filler content that optimizes the
delivered capacity of the battery and being that delivery capacity independent of the conductive filler content for higher
concentrations.
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8.
Carbohydrate counting: how accurate should it be to achieve glycemic control in patients on intensive insulin regimens?
Abreu, Carlos and Miranda, Francisco and Felgueiras, Paula
AIP Conference Proceedings
AIP Publishing
Carbohydrate counting is an important mealplanning tool for patients on intensive insulin regimens. Preprandial insulin bolus is adjusted taking into account the carbohydrate content of each meal and the insulintocarb ratio of each patient throughout the day. Evidence suggests that accurate carbohydrate counting may have positive effects not only on reducing glycosylated hemoglobin concentration but also on decreasing the incidence of hypoglycemic episodes. Nevertheless, despite its benefits, the efficacy of carbohydrate counting depends on the ability of each patient, or its caregiver, to accurately estimate the carbohydrate content of each meal. Therefore, it is of great importance to understand how accurate should carbohydrate counting be, and the impact of inaccurate carbohydrate counting on the glycemic control of each patient. Within this work, we propose an analytic method that uses the insulintocarb ratio and the insulin sensitivity factor, along with the glycemic targets of each patient to calculate the limits of accurate carbohydrate counting, in order to achieve better glycemic control and to reduce hypoglycemic episodes.
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7.
Home glucose meters: how accurate should they be to avoid dysglycemia in patients using carbohydrate counting?
Abreu, Carlos and Miranda, Francisco and Dabrowska, Anna and Felgueiras, Paula
AIP Conference Proceedings
AIP Publishing
Accurate selfmonitoring of blood glucose is the key to an effective and safe intensive insulin therapy. Indeed, most insulin dosing decisions are made based on the blood glucose values obtained from home glucose meters, in particular for those using diet planning and carbohydrate counting. Patients on that therapeutic regimen depend not only on their ability to accurately
estimate the carbohydrate content of each meal but also on the accuracy of the glucose meter being used. Therefore, in order to avoid postprandial hypoglycemia and hyperglycemia, it is of great importance to realize how important is the accuracy of blood glucose meters according to the particular characteristics of each patient. In this regard, we propose an analytic method to find the limits of the blood glucose meters accuracy according to the insulintocarb ratio, the insulin sensitivity factor and the ability of each patient to estimate the carbohydrate content of each meal.
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6.
An adaptive mealtime bolus calculator to minimize the effects of inaccurate carbohydrate counting
Abreu, Carlos and Miranda, Francisco and Felgueiras, Paula
AIP Conference Proceedings
AIP Publishing
Evidence suggests that accurate carbohydrate counting along with selfmonitoring of blood glucose is the key to a successful diabetes management, in particular for patients on intensive insulin regimens. However, despite its benefits, accurate
carbohydrate counting is a complex, difficult, timeconsuming, and errorprone task for most patients. Several studies show that
most patients frequently estimate the carbohydrate content of meals within an error of about 1015 g of the real value. In addition, fearing hypoglycemic events, patients frequently underestimate the carbohydrate content of meals and, consequently, they have high levels of HbA1C. Therefore, is important to avoid the consequences of incorrect carbohydrate counting in order to improve the patient’s glycemic control. To that end, this work presents an adaptive mealtime bolus calculator that uses the patient’s glycemic data to dynamically adjust the mealtime bolus and counterbalance the negative effects of inaccurate carbohydrate counting.
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5.
A fractional measles model having monotonic real statistical data for constant transmission rate of the disease
Almeida, Ricardo and Qureshi, Sania
Fractal and Fractional
MDPI
NonMarkovian effects have a vital role in modeling the processes related with
natural phenomena such as epidemiology. Various infectious diseases have longrange memory
characteristics and, thus, nonlocal operators are one of the best choices to be used to understand the
transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order
epidemiological model of measles. Some relevant features, such as wellposedness and stability of
the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically
stable equilibrium point for basic reproduction number R0 < 1, which is most sensitive to the
fractional order parameter and to the percentage of vaccination. We show the efficiency of the model
through a real life application of the spread of the epidemic in Pakistan, comparing the fractional
and classical models, while assuming constant transmission rate of the epidemic with monotonically
increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type
model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals
when compared with the classical model.
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4.
Optimal control of HIV treatment and immunotherapy combination with state and control delays
Silva, Cristiana J. and Maurer, Helmut
Optimal Control Applications and Methods
Wiley
In this paper, we propose and analyze an optimal control problem where human immunodeficiency virus treatment and immunotherapy are described by two control functions that are subject to time delays representing pharmacological and absorption delays, respectively. The goal is to propose effective optimal control solutions for the combination of human immunodeficiency virus treatment and immunotherapy, ensuring a functional behavior of the immune system. The incubation period is mathematically represented by a time delay in the virus load, and the local asymptotic and Hopf bifurcation analysis of the CTLequilibrium point of the uncontrolled delayed system is studied. We obtain optimal controls of bangsingular type both for the nondelayed and delayed optimal control problem with and without state constraints.We study boundary arcs of state constraints and junction properties of the control and adjoint variables at entry and exit points of boundary arcs. Moreover, we derive an explicit formula of the multiplier associated with the state constraint.
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3.
Further properties of Osler's generalized fractional integrals and derivatives with respect to another function
Almeida, Ricardo
Rocky Mountain Journal of Mathematics
Rocky Mountain Mathematics Consortium
In this paper we discuss fractional integrals
and fractional derivatives of a function with respect to another function. We present some fundamental properties for
both types of fractional operators, such as Taylor’s theorem,
Leibniz and semigroup rules. We also provide a numerical
tool to deal with these operators, by approximating them
with a sum involving integerorder derivatives.
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2.
A stochastic analysis of the impact of fluctuations in the environment on preexposure prophylaxis for HIV infection
Djordević, Jasmina and Silva, Cristiana J.
Soft Computing
Springer Nature
We propose a stochastic model for HIV/AIDS transmission where preexposure prophylaxis is considered as a prevention measure for new HIV infections. A white noise is introduced into the model, representing fluctuations in the environment that manifest themselves on the transmission coefficient rate. We prove the existence and uniqueness of a global positive solution of the stochastic model and establish conditions under which extinction and persistence in mean hold. Numerical simulations are provided which illustrate the theoretical results, and conclusions are derived on the impact of the fluctuations in the environment on the number of the susceptible individuals that are under preexposure prophylaxis.
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Proceedings
1.
The impact of accurate carbohydrate counting on patient’s glycemic targets
Abreu, C. and Miranda, F. and Felgueiras, P.
Diabetes Technology and Therapeutics
Mary Ann Liebert
Preprandial insulin bolus is adjusted taking into account the carbohydrate content of each meal, the patient’s glycemic targets (GHyper, GT and GHypo), the insulin sensitivity factor (ISF), and the insulintocarb ratio (ICR) throughout the day. Evidence suggests that accurate carbohydrate counting may have positive effects not only on reducing glycosylated hemoglobin concentration but also on decreasing the incidence of hypoglycemic episodes. Therefore, the efficacy of carbohydrate counting depends not only on the ability of each patient accurately estimate the carbohydrate content of each meal but also on each patient glycemic targets.
ria.ua.pt

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Peer Reviewed