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Tatiana Odzijewicz

Publications and source records attributed to Tatiana Odzijewicz.

At least 19 recordsLinked to original sources
arXiv2017

Non-invasive control of the fractional Hegselmann-Krause type model

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.

math.DS↗
arXiv2016

A remark on local fractional calculus and ordinary derivatives

In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

math.CA↗
arXiv2016

Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo-Katugampola derivative. The Caputo and the Caputo-Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo--Katugampola derivative, is proven. A decomposition formula for the Caputo-Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

math.CA↗
arXiv2016

Variational methods for the solution of fractional discrete/continuous Sturm-Liouville problems

The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method.

math.NA↗
arXiv2014

Generalized Fractional Calculus of Variations

In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In particular, we obtain necessary optimality conditions of Euler-Lagrange type for the fundamental and isoperimetric problems, natural boundary conditions, and Noether theorems. Existence of solutions is shown under Tonelli type conditions. Moreover, we apply our results to prove existence of eigenvalues, and corresponding orthogonal eigenfunctions, to fractional Sturm-Liouville problems.

math.OC↗
arXiv2014

Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition --- Application to fractional variational problems

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. Main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].

math.OC↗
arXiv2014

The Generalized Fractional Calculus of Variations

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental and isoperimetric problems, natural boundary conditions, and Noether type theorems.

math.OC↗
arXiv2013

Fractional Calculus of Variations of Several Independent Variables

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given.

math-ph↗
arXiv2013

Variational Methods for the Fractional Sturm--Liouville Problem

This article is devoted to the regular fractional Sturm--Liouville eigenvalue problem. Applying methods of fractional variational analysis we prove existence of countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.

math.OC↗
arXiv2013

A Generalized Fractional Calculus of Variations

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.

math.OC↗
arXiv2013

Noether's theorem for fractional variational problems of variable order

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

math.OC↗
arXiv2012

Noether's Theorem for Nonsmooth Extremals of Variational Problems with Time Delay

We obtain a nonsmooth extension of Noether's symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler-Lagrange extremals are restricted to those that satisfy the DuBois-Reymond necessary optimality condition. The important case of delayed variational problems with higher-order derivatives is considered as well.

math.OC↗
arXiv2012

Green's Theorem for Generalized Fractional Derivatives

We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.

math.CA↗
arXiv2012

Variable Order Fractional Variational Calculus for Double Integrals

We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional Euler-Lagrange necessary optimality condition for variable order two-dimensional fractional variational problems.

math.OC↗
arXiv2012

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.

math.OC↗
arXiv2012

Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to Physics discussed.

math.OC↗
arXiv2012

Calculus of Variations with Classical and Fractional Derivatives

We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental problem of the calculus of variations with mixed integer and fractional order derivatives as well as isoperimetric problems are considered.

math.OC↗