Related papers: Fractional Variational Calculus with Classical and…
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…
Fractional operators play an important role in modeling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of…
We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a…
We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and…
Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…
An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the…
We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we…
In this paper, we investigate a fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Using techniques inspired by earlier works on…
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…
We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional…
We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
We set the main concepts for multiplicative fractional calculus. We define Caputo, Riemann and Letnikov multiplicative fractional derivatives and multiplicative fractional integrals and study some of their properties. Finally, the…
This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…
In this paper we study isoperimetric problems of the calculus of variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with…
In this paper we study linear and nonlinear fractional differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<\alpha<1.$ We first obtain a new estimate of the fractional…
In this article, we employ a fractional version of the radius of curvature in Euler's equation for column buckling, enabling us to derive a fractional differential equation in the Caputo sense. We solve this equation and demonstrate that…
The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally…