An analytical solution of the fractal toda oscillator

Abstract

In this paper, the fractal Toda oscillator is established by using the fractal variational theory and the exact analytical solution is obtained by the non-perturbative method. Also, trigonometric series and He’s frequency formula are applied to determine the approximate analytical solution. To illustrate the operability and effectiveness of these methods, the solution processes are described in detail, and the analytical solutions are compared with numerical ones, showing good agreement. It should be noted that the relative errors produced by He's frequency formula and the trigonometric series method are all small. The results show that the non-perturbative method and He's frequency formula are extremely effective and simple for fractal differential equations. Finally, the effect of fractal derivative order on the vibration characteristics is briefly explained graphically. The non-perturbative method provides additional advantages for quickly testing the frequency property of fractal nonlinearity vibration, which is of great significance in the study of the vibration characteristics of nonlinear oscillators.

Keywords

• The Toda oscillator
• Non-perturbative method
• Trigonometric series
• He’s fractal frequency formula
• The fractal variational theory
• Two-scale transform method