Core Concepts

Derivation of higher-order fractional variational integrators using convolution quadrature.

Abstract

The content discusses the derivation of fractional variational integrators based on convolution quadrature for dissipative systems. It covers the theoretical framework, numerical methods, convergence properties, and applications to specific problems like the damped oscillator and the Bagley-Torvik equation. The focus is on developing higher-order integrators for fractional dynamics.
Introduction to Fractional Calculus
Fractional calculus extends classical integration and differentiation.
Non-locality in fractional calculus models various physical phenomena.
Discrete Hamilton's Principle
Discretization of Lagrangian systems using variational integrators.
Derivation of discrete Euler-Lagrange equations for forced systems.
Fractional Integrals and Derivatives
Definition and properties of fractional integrals and derivatives.
Convolution quadrature as a numerical tool for approximating fractional integrals.
Restricted Variational Principle
Introduction of a restricted variational principle for Lagrangian systems with fractional damping.
Formulation of equations for extremals under restricted variations.
Numerical Experiment
Application of fractional variational integrators to the damped harmonic oscillator and the Bagley-Torvik equation.
Analysis of energy dissipation, absolute errors, and global errors for different integrator orders.
Higher-Order Integrators
Development of higher-order fractional variational integrators using convolution quadrature.

Stats

Fractional derivative operator: Dα = J−α
Lagrangian for harmonic oscillator: L(x, ˙x) = ˙x^2/2 - x^2/2
Bagley-Torvik equation: ¨x + µD(2α)−x + x = f(t)

Quotes

"The construction of the desired forced variational integrators was motivated by previous work."
"Convolution quadrature preserves structure in the fractional framework."

Deeper Inquiries

The convergence order of variational integrators plays a crucial role in determining the accuracy of numerical simulations. A higher convergence order implies that the numerical method can approximate the solution more accurately with fewer computational resources. This means that as the convergence order increases, the error in the numerical solution decreases at a faster rate. Therefore, higher-order variational integrators are preferred for achieving more precise results in numerical simulations. Additionally, a higher convergence order allows for the use of larger time steps without sacrificing accuracy, leading to faster computational times.

Using convolution quadrature for approximating fractional integrals has several practical implications. Firstly, convolution quadrature preserves the structure of fractional operators, such as integration by parts and semigroup properties, which are essential for maintaining the accuracy and stability of numerical methods. Additionally, convolution quadrature allows for the approximation of fractional integrals with high accuracy, enabling the efficient numerical simulation of systems with fractional derivatives. This method is particularly useful for studying non-local physical phenomena and damping models, where fractional calculus is applied to model complex dynamics accurately.

The concept of variational integrators can be applied to various areas of physics and engineering beyond the examples discussed in the article. For instance, in computational fluid dynamics, variational integrators can be used to simulate fluid flow and turbulence accurately. In structural mechanics, variational integrators can model the behavior of complex structures under dynamic loads. In control systems engineering, variational integrators can be employed to design and analyze control algorithms for robotic systems and autonomous vehicles. Overall, variational integrators provide a versatile and powerful numerical framework for simulating a wide range of physical systems with high accuracy and efficiency.

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