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Title: Exact Solution for Nonlinear Oscillators with Coordinate-Dependent Mass
Authors: Abu-As'ad, Ata
Asad, Jihad
Keywords: Homotopoty;Perturbation;Equilibrium (fixed) points;Simulation;stability,;Exact differential Equation;potential function;Conservative;Reduction of order;Center point
Issue Date: 22-Apr-2022
Citation: Abu- As’ad, A., & Asad, J. (2022). Exact Solution for Nonlinear Oscillators with Coordinate-Dependent Mass. European Journal of Pure and Applied Mathematics, 15(2), 496–510.
Series/Report no.: Vol. 15, No. 2;496-510
Abstract: In this work, we aim to obtain an exact solution for a nonlinear oscillator with coordinate position- dependent mass. The equation of motion of the nonlinear oscillator under investigation becomes exact after making reduction of order. The obtained solution was expressed in terms of position and time. Initial conditions were applied, in addition to modified initial condition. Finally, fixed points where studied with their stability, and some plots describing the system where presented.
Description: Simple oscillating systems are modeled in general as a mass attached to a spring (i.e., simple oscillators). The equation of motion describing such systems are obtained either using Newtonian mechanics or Lagrangian method, and it can be solved exactly in some simple cases. Unfortunately, no such systems present in the macroscopic world and this is due to dissipative forces that are always present in nature. Dissipative forces can be ignored if they have small effects, but in many cases they lead to damping oscillators. Linear oscillators are those that oscillate with one frequency and its motion is sinusoidal and periodic, for more information related to oscillators (simple and damped) we advise interested people to refer to some classical mechanics texts [5, 9, 10]. Nonlinear oscillators result in complex motion and there are mainly two important features for such systems: as the amplitude increases then the non linearity motion becomes more important, and in some cases, the frequency will change with amplitude. In real world one can find many such nonlinear oscillators and one has to note that coupled nonlinear oscillators are a subject founded in many branches of science as: biology, physics, and many others. In literature there are a lot of efforts paid on studying these systems [13, 16, 20]. An important example is the van der Pol oscillator which is an oscillator with nonlinear damping introduced in the 1920’s by Balthasar van der Pol (1889 - 1959).The van der Pol oscillator is considered as an example of an oscillator with nonlinear damping, energy being dissipated at large amplitudes and generated as low amplitude, and it attracts the attention of many researchers where many method have been applied in dealing with this oscillator either analytically using Homotopy analysis method (HAM) as in[4, 14]. Homotopy perturbation method (HPM) as in [18] or numerically using for example perturbation algorithm combining the method of Multiple Scales and Modified Lindstedt–Poincare Techniques as in [15], A domain decomposition method (ADM) as in [2, 8] and many other methods. Nonlinear oscillations have been of paramount importance in practical engineering, physics, applied mathematics, and several real-world requirements for many years. In literature, one can find many various analytical approaches for solving nonlinear systems, such as the iteration perturbation method [7], the homotopy perturbation method (HMP) [21], the variational method [17], and many other methods[6]. Interested researchers in this topic can refer to [12, 12] therein. In principle, the solution for such nonlinear oscillators is difficult to obtain analytically and researchers resort to use different numerical methods [1, 3, 21, 21]. In [21] the authors consider a nonlinear oscillator with coordinate-dependent mass, where they proposed a nonlinear oscillator with negative coefficient of linear term (see Eq. 3 in [21]) and apply the homotopy perturbation method to find an approximate period for their equation. In this paper we are going to find an exact solution for the above equation in section 2, while in section 3 the exact solution with modifying in the potential function will be presented and explained, an equilibrium points of the system and their stability with graphical simulation are given finally close the paper with a conclusion .
ISSN: 1307-5543 –
Appears in Collections:Applied science faculty

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