87 research outputs found
Higher-Order Calculus of Variations on Time Scales
We prove a version of the Euler-Lagrange equations for certain problems of
the calculus of variations on time scales with higher-order delta derivatives.Comment: Corrected minor typo
First- and second-order dynamic equations with impulse
We present existence results for discontinuous first- and continuous second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions
Towards the Formalization of Fractional Calculus in Higher-Order Logic
Fractional calculus is a generalization of classical theories of integration
and differentiation to arbitrary order (i.e., real or complex numbers). In the
last two decades, this new mathematical modeling approach has been widely used
to analyze a wide class of physical systems in various fields of science and
engineering. In this paper, we describe an ongoing project which aims at
formalizing the basic theories of fractional calculus in the HOL Light theorem
prover. Mainly, we present the motivation and application of such formalization
efforts, a roadmap to achieve our goals, current status of the project and
future milestones.Comment: 9 page
Direct and Inverse Variational Problems on Time Scales: A Survey
We deal with direct and inverse problems of the calculus of variations on
arbitrary time scales. Firstly, using the Euler-Lagrange equation and the
strengthened Legendre condition, we give a general form for a variational
functional to attain a local minimum at a given point of the vector space.
Furthermore, we provide a necessary condition for a dynamic
integro-differential equation to be an Euler-Lagrange equation (Helmholtz's
problem of the calculus of variations on time scales). New and interesting
results for the discrete and quantum settings are obtained as particular cases.
Finally, we consider very general problems of the calculus of variations given
by the composition of a certain scalar function with delta and nabla integrals
of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be
published in the Springer Volume 'Modeling, Dynamics, Optimization and
Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer
Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted,
after a revision, 19/Jan/201
Generalized transversality conditions for the Hahn quantum variational calculus
We prove optimality conditions for generalized quantum variational problems
with a Lagrangian depending on the free end-points. Problems of calculus of
variations of this type cannot be solved using the classical theory
- …