Related papers: The Delta-nabla Calculus of Variations for Composi…
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…
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…
We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler-Lagrange…
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on…
We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.
The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the…
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.
We introduce the differential, integral, and variational delta-embeddings. We prove that the integral delta-embedding of the Euler-Lagrange equations and the variational delta-embedding coincide on an arbitrary time scale. In particular, 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…
We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale…
The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler-Lagrange type for the basic, isoperimetric, and Lagrange variational problems are…
The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing…
We consider geometric variational problems for a functional defined on a curve in three-dimensional space. The functional is assumed to be written in a form invariant under the group of Euclidean motions. We present the Euler-Lagrange…
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 derive Euler-Lagrange type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order $(\alpha,\beta)$, $\alpha > 0$, $\beta > 0$, recently introduced by J.…
We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional…
We establish Euler-Lagrange equations for a problem of Calculus of variations where the unknown variable contains a term of delay on a segment.
We prove a Noether-type symmetry theorem and a DuBois-Reymond necessary optimality condition for nabla problems of the calculus of variations on time scales.
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.