Analysis of Vibrations in Large Flexible Hybrid Systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions results in the impossibility to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameter together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different character take place, are defined. Dependence of the vibration frequencies on physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allow one to control the frequency spectrum, in which excitation of vibrations is possible.Comment: Accepted for publication by the Global Journal of "Pure and Applied Mathematics". To be partially presented at the Sixth International Conference "Symmetry in Nonlinear Mathematical Physics", June 20-26, 2005, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (Kiev), Ukrain

An integral boundary fractional model to the world population growth

We consider a fractional differential equation of order $\alpha$, $\alpha \in (2,3]$, involving a $\psi$-Caputo fractional derivative subject to initial conditions on function and its first derivative and an integral boundary condition that depends on the unknown function. As an application, we investigate the world population growth. We find an order $\alpha$ and a function $\psi$ for which the solution of our fractional model describes given real data better than available models.The authors are supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT).publishe

A Maple interface for computing variational symmetries in optimal control

A computer algebra package, for the automatic computation of variational symmetries in optimal control, was recently developed by the authors [2,3]. Now we present a graphical user interface which permit to interact, in a point-and-click environment, with all the previous symbolical tools

Computation of conservation laws in optimal control

Making use of a computer algebra system, we define computational tools to identify symmetries and conservation laws in optimal control

Uma forma bidimensional que maximiza a resistência aerodinâmica newtoniana

In a previous work [18, 19] it is investigated, by means of computational simulations, shapes of nonconvex bodies that maximize resistance to its motion on a rare ed medium, considering that bodies are moving forward and at the same time slowly rotating. Here the previous results are improved: we obtain a two-dimensional geometric shape that confers to the body a resistance very close to the supremum value (R = 1:4965 < 1:5). Um corpo bidimensional, apresentando um ligeiro movimento rotacional, desloca-se num meio rarefeito de partículas que colidem com ele de uma forma perfeitamente elástica. Em investigações que os dois primeiros autores realizaram anteriormente [18, 19], procuraram-se formas de corpos que maximizassem a força de travagem do meio ao seu movimento. Dando continuidade a esse estudo, encetam-se agora novas investigações que culminam num resultado que representa um grande avanço qualitativo relativamente aos então alcançados. Esse resultado, que agora se apresenta, consiste numa forma bidimensional que confere ao corpo uma resistência muito próxima do seu limite teórico. Mas o seu interesse não se fica pela maximização da resistência newtoniana; atendendo às suas características, apontam-se ainda outros domínios de aplicação onde se pensa poder vir a revelar-se de grande utilidade. Tendo a forma óptima encontrada resultado de estudos numéricos, é objecto de um estudo adicional de natureza analítica, onde se demonstram algumas propriedades importantes que explicam em grande parte o seu virtuosismo

Uma forma bidimensional que maximiza a resistência aerodinâmica newtoniana

In a previous work [18, 19] it is investigated, by means of computational simulations, shapes of nonconvex bodies that maximize resistance to its motion on a rare ed medium, considering that bodies are moving forward and at the same time slowly rotating. Here the previous results are improved: we obtain a two-dimensional geometric shape that confers to the body a resistance very close to the supremum value (R = 1:4965 < 1:5). Um corpo bidimensional, apresentando um ligeiro movimento rotacional, desloca-se num meio rarefeito de partículas que colidem com ele de uma forma perfeitamente elástica. Em investigações que os dois primeiros autores realizaram anteriormente [18, 19], procuraram-se formas de corpos que maximizassem a força de travagem do meio ao seu movimento. Dando continuidade a esse estudo, encetam-se agora novas investigações que culminam num resultado que representa um grande avanço qualitativo relativamente aos então alcançados. Esse resultado, que agora se apresenta, consiste numa forma bidimensional que confere ao corpo uma resistência muito próxima do seu limite teórico. Mas o seu interesse não se fica pela maximização da resistência newtoniana; atendendo às suas características, apontam-se ainda outros domínios de aplicação onde se pensa poder vir a revelar-se de grande utilidade. Tendo a forma óptima encontrada resultado de estudos numéricos, é objecto de um estudo adicional de natureza analítica, onde se demonstram algumas propriedades importantes que explicam em grande parte o seu virtuosismo

Helmholtz Theorem for nondifferentiable Hamiltonian systems in the framework of Cresson's quantum calculus

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson's quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian. © 2016 Frédéric Pierret and Delfim F. M. Torres

A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.Comment: Accepted for publication in the Journal of Mathematical Analysis and Application

Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales

We introduce the concept of fractional derivative of Riemann-Liouville on time scales. Fundamental properties of the new operator are proved, as well as an existence and uniqueness result for a fractional initial value problem on an arbitrary time scale. © 2015 The Authors

Noether's Theorem on Time Scales

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.Comment: Partially presented at the 6th International ISAAC Congress, August 13 to August 18, 2007, Middle East Technical University, Ankara, Turke