Abstract
The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically.
Highlights
Method for the Fractal TodaAn oscillation occurs when its kinetic energy and its potential energy are changed alternatively, while the total energy remains unchanged
The nonlinear oscillator can be obtained from Equation (8) as a stationary condition:
The fractal variational formulation given in Equation (11) becomes: Z
Summary
An oscillation occurs when its kinetic energy and its potential energy are changed alternatively, while the total energy remains unchanged. Its variational formulation can be expressed as [1,2,3]: Z. A nonlinear oscillator can be written in the form: iations. Du dt with the initial conditions: Licensee MDPI, Basel, Switzerland. For an oscillator, it requires [5,6]: distributed under the terms and conditions of the Creative Commons. The nonlinear oscillator can be obtained from Equation (8) as a stationary condition:. This is the Toda oscillator [7,8,9,10]
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