Two analytical methods for time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

Abstract

This paper focuses on solving the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation (FCDGSKE). We propose two analytical methods based on the fractional complex transform, the variational iteration method and the homotopy perturbation method. The approximated solutions to the initial value problems associated with FCDGSKE are provided without linearization and complicated calculation. Numerical results show the main merits of the analytical approaches.

Numerical investigation of the fractal capillary oscillator

“…This approach was first proposed by Lu and Chen for the numerical analysis of the fractal Yao-Cheng oscillator in a fractal space. 14 TST-GRHBM consists of two steps, the fractal nonlinear oscillator is first represented as a classical nonlinear oscillator by the two-scale transformation (TST) proposed by He 20–26 and the approximations for the transformed oscillator can be obtained by the global residue harmonic balance method (GRHBM). 14,27–30 For illustrating the procedure of TST-GRHBM for (2), we first transform the fractal capillary oscillator as the classical capillary oscillator by using the two-scale fractal transformation.…”

“…The difficulty for solving the fractal capillary oscillator (2) lies in two sides, where one is the fractal operator defined by He’s derivative and the other is that how many nonlinear approximated terms of sin( u ) are required for different amplitudes A . For releasing the fractal operator in (2), we consider the two-scale transformation 20–26 and transform the original fractal capillary oscillator as the classical capillary oscillator. By using the two-scale transformation given by T = t α , we have the following capillary oscillator 7,8 with the transformed initial conditions as follows…”

“…We remark that the physical explanation of the two-scale transformation was given in the references. 21–26 For two adjacent hierarchical levels of a Carton set, 21 when we measure it by using a large-scale, it is a continuous line, and when we consider it on a small scale, it becomes discontinuous.…”