arXiv subjects

Fernando Olivar-Romero

Publications and source records attributed to Fernando Olivar-Romero.

5 recordsLinked to original sources

arXiv2026

Fractional oscillators with fractional damping in the presence of external forces

We investigate how a fractional oscillator reacts to external forces when the dynamic law includes a damping term that is explicitly fractional. The corresponding three-parametric fractional differential equation (in the Caputo sense) admits exact solution. Two of the fractional parameters are associated with the intrinsic dissipation mechanism that produces continuous dissipation of energy. The third fractional parameter characterizes the damping term against which the oscillator is driven by the external force. When external time-dependent forces are included, the fractional dynamic law is associated with the Newton-Scott-Blair model of viscoelastic materials. The Laplace transform of the external force defines the profile of the solution; specific examples include the absence of external forces as well as constant, sinusoidal and stepped external forces. With a constant driving force, the system reaches a fixed position after a transient period. With a sinusoidal driving force, the system exhibits persistent oscillatory behavior. These results suggest that the external driving force determines the long-term behavior of the fractionally damped oscillator.

math-ph↗

arXiv2019

An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

We solve the Cauchy problem defined by the fractional partial differential equation $[\partial_{tt}-κ\mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions $u(x,0)=μ(\sqrtπx_0)^{-1}e^{-(x/x_0)^2}$, $u_t(x,0)=0$. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse $u(x,0)$ by the Dirac delta distribution $φ(x)=μδ(x)$ is obtained as corollary.

math-ph↗

arXiv2018

Transition from the wave equation to either the heat or the transport equations through fractional differential expressions

We study a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that nonlocal differential expressions are required to describe the time-evolution of solutions. Non-locality is modeled with a space-time fractional differential equation of order $1 \leq α\leq 2$ in time, and order $1\leq β\leq 2$ in space. We adopt the notion of Caputo for time-derivative and the Riesz pseudo-differential operator for space-derivative. The corresponding Cauchy problem is solved for zero initial velocity and initial disturbance represented by either the Dirac delta or the Gaussian distributions. Well known results for the partial differential equations of wave propagation, diffusion and (modified) transport processes are recovered as particular cases. In addition, regular solutions are found for the partial differential equation that arises from $α=2$ and $β=1$. Unlike the above cases, the latter equation permits the presence of nodes in its solutions.

math-ph↗

arXiv2017

Fractional Driven Damped Oscillator

The resonances associated with a fractional damped oscillator which is driven by an oscillatory external force are studied. It is shown that such resonances can be manipulated by tuning up either the coefficient of the fractional damping or the order of the corresponding fractional derivatives.

physics.class-ph↗

arXiv2016

Factorization of the Quantum Fractional Oscillator

The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the `factorization energy' is now a fractional-differential operator rather than a constant. As a first example, the energies and wave-functions of a fractional version of the quantum oscillator are determined. Interestingly, the energy eigenvalues are expressed as power-laws of the momentum in terms of the non-integer differential order of the eigenvalue equation.

math-ph↗