Models and Numerical Solutions of Generalized Oscillator Equations

Abstract

In this paper, we use three operators called K, A, and Boperators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K, A, and Boperators allow the kernel to be arbitrary. In the case, when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K, A, and Boperators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler–Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A finite difference scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution, which demonstrates that the numerical scheme is convergent.