Stability analysis of fractional differential equations with unknown parameters

In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. Using the graphical based D-decomposition method, the parametric stability analysis of FDEs is investigated without complicated mathematical analysis. To achieve this, stability boundaries are obtained firstly by a conformal mapping from s-plane to parameter space composed by unknown parameters of FDEs, and then the stability region set depending on the unknown parameters is found. The applicability of the presented method is shown considering some benchmark equations, which are often used to verify the results of a new method. Simulation examples show that the method is simple and give reliable stability results. &nbsp

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time 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 delta-integral 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.Comment: This is a preprint of a paper accepted for publication as a book chapter with Springer International Publishing AG. Submitted 23/Jan/2019; revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial text overlap with arXiv:1508.0075

A New Efficient Technique for Solving Modified Chua's Circuit Model with a New Fractional Operator

Chua's circuit is an electronic circuit that exhibits nonlinear dynamics. In this paper, a new model for Chua's circuit is obtained by transforming the classical model of Chua's circuit into novel forms of various fractional derivatives. The new obtained system is then named fractional Chua's circuit model. The modified system is then analyzed by the optimal perturbation iteration method. Illustrations are given to show the applicability of the algorithms, and effective graphics are sketched for comparison purposes of the newly introduced fractional operatorsThe authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-1

Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

Fractional Calculus and Special Functions with Applications

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications

Conformable Derivative Operator in Modelling Neuronal Dynamics

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense

Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integro-differential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We here study the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time averaged mean squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integro-differential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integro-differential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.Comment: 26 pages, 7 figures, RevTe

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

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 Operators and the Mittag-Leffler Function

This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others