Numerical Solutions of Generalized Oscillator Equations

Abstract

Harmonic oscillators play a fundamental role in many areas of science and engineering, such as classical mechanics, electronics, quantum physics, and others. As a result, harmonic oscillators have been studied extensively. Classical harmonic oscillators are defined using integer order derivatives. In recent years, fractional derivatives have been used to model the behaviors of damped systems more accurately. In this paper, we use three operators called K-, A- and B-operators 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 B-operators 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 B-operators, 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 numerical 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. It is demonstrated that the numerical scheme is convergent, and the order of convergence is 2. For a special kernel, this scheme reduces to a scheme presented recently in the literature.