In this paper a fractional differential equation of the Euler-Lagrange/ Sturm-Liouville type is considered. The fractional equation with derivatives of order α ∈ (0, 1] in the finite time interval is transformed to the integral form. Next the numerical scheme is presented. In the final part of this paper examples of numerical solutions of this equation are shown. The convergence of the proposed method on the basis of numerical results is also discussed.MSC 2010 : Primary 26A33; Secondary 34A08, 65L10

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.

In this paper we have examined the stable and unstable states of the operation during the emptying of the silo. Here the state of stable operation of the silo is understood as a smooth mass outflow. The unstable state is manifested by funnel flow or arching. This issues from a moistured granular material. We have focused on the two extreme experimental cases. The first option was considered as a mass flow and the second one was considered as arching. Based on experimental data we have simulated the silo emptying by DEM. Considering the transition between stable and unstable operation we proposed a novel mathematical model of the silo emptying. This model involves a fractional-differential oscillator equation. Analyzing the solution of this equation we have presented how to recognize the state of stable and unstable operation during silo emptying.