In this study, we consider two coupled pendulums (attached together with a spring) having the same length while the same masses are attached at their ends. After setting the system in motion we construct the classical Lagrangian, and as a result, we obtain the classical Euler-Lagrange equation. Then, we generalize the classical Lagrangian in order to derive the fractional Euler-Lagrange equation in the sense of two different fractional operators. Finally, we provide the numerical solution of the latter equation for some fractional orders and initial conditions. The method we used is based on the Euler method to discretize the convolution integral. Numerical simulations show that the proposed approach is efficient and demonstrate new aspects of the real-world phenomena.

Abstract

In the present work, the lattice Green’s function technique has been used to investigate the equivalent two-site resistance between arbitrary pairs of lattice sites in infinite, generalized decorated square and simple cubic lattices with identical resistors. Some results for the resistance are presented. The results for the generalized decorated square lattice are numerically confirmed by commercial software (National Instruments software Multisim). The asymptotic values of the resistance for the generalized decorated simple cubic lattice are calculated numerically when the separation of the two lattice points goes to infinity