The Motion of a Bead Sliding on a Wire in Fractional Sense

Abstract

In this study, we consider the motion of a bead sliding on a wire which is bent into a parabola form. We first introduce the classical Lagrangian from the system model under consideration and obtain the classical EulerLagrange equation of motion. As the second step, we generalize the classical Lagrangian to the fractional form and derive the fractional Euler-Lagrange equation in terms of the Caputo fractional derivatives. Finally, we provide numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on a discretization scheme using a Grünwald-Letnikov approximation for the fractional derivatives. Numerical simulations verify that the proposed approach is efficient and easy to implement.

Projective synchronization of fractional-order delayed neural networks based on the comparison principle

“…In the control scheme (21), (22), (23), (24), we select the symmetric positive definite matrix P = Similarly, projective synchronization with projective coefficient β = -2 is shown in Fig. 12-Fig.…”

“…Compared to classical integer-order models, fractional-order calculus offers an excellent instrument for the description of memory and hereditary properties of dynamical processes. The existence of infinite memory can help fractional-order models better describe the system's dynamical behaviors as illustrated in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Taking these factors into consideration, fractional calculus was introduced to neural networks forming fractional-order neural networks, and some interesting results on synchronization were demonstrated [24][25][26][27][28][29].…”

Duhamel's formula for time‐fractional Schrödinger equations

“…1,2 On the other hand, fractional calculus has gained much popularity due to their extensive applications in a variety of fields such as physics, mechanics, chemistry, and engineering. The recent advancements in fractional derivatives and their applications can be found in other papers [3][4][5][6][7][8][9][10][11] and the references therein.…”

An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

“…Recently, a series of definition of the fractional derivative were proposed [4][5][6]. These new definitions can better describe the chemical kinetics system pertaining [7], the generation of nonlinear waterwaves in the long-wavelength regime [8], the convective straight fins with temperaturedependent thermal conductivity [9], the relaxation and diffusion models [4], optimal con-trol problems [5], the motion of a bead sliding on a wire [10], the material heterogeneities and structures with different scales [6].…”