Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations

Abstract

In this work, the variational iteration algorithm-I with an auxiliary parameter is used for the analytical treatment of the wave equations and wave-like vibration equations. The technique has the capability of reducing the size of computational work and easily overcomes the difficulty of the perturbation method or Adomian polynomials. Comparison with the classic variational iteration algorithm-I (VIA-1) is carried out, showing that the modification is more efficient and reliable. Keywords Variational iteration algorithm-I, wave equation, wave-like vibration equation, VIA-I with an auxiliary parameter where g(u) is a linear function. The vibrational behavior of beams and shafts can be expressed in terms of waves. 1 The motion of the system of vibrating beams, rods, and springs reduces to that of the wave equation. The wave equation is probably the most used partial differential equation (PDE) in practical applications. Waves exist in different media like as in geophysics there are surface and internal waves in the ocean, but there are also more complex nonlinear models comparing to PDE. There are wave processes in the Earth (seismic waves), acoustic waves in the air and water, electromagnetic waves in different media. Initial equations differ for each case, but the wave equation can be derived in some approximation in most of the cases.

The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators

“…Nonlinear oscillations arise everywhere in our everyday life and engineering. As an exact solution might be too complex to be used for a practical application, many analytical methods have been used in open literature, for example, the variational iteration method, [1][2][3][4][5][6][7] the homotopy perturbation method, [8][9][10][11][12][13][14][15][16][17][18][19][20] He-Laplace method, [21][22][23] the variational approach [24][25][26][27][28][29] and the Hamiltonian approach. 30,31 The most important property of a nonlinear oscillator is the relationship between the frequency and its amplitude, the simplest method to estimate the frequency-amplitude relationship might be He's frequency formulation [32][33][34] and the max-min approach, 35,36 which are still under development and many modifications were proposed to improve the accuracy.…”

Analytical solutions to contact problem with fractional derivatives in the sense of Caputo

“…The VIM was presented by Inokuti et al [30]. Moreover, the approach introduced by He [31] has been applied in numerous fields of applied and pure science for addressing a wider range of problems [32,33]. For fast convergence of solution, selction of initial approximation is very important in VIM, for details, see [34].…”

Numerical solution of second order Painlevé differential equation

“…The estimated solution u k (x, h) has the helper parameter h, which guarantees the intermingling to the precise solution. This procedure is known as variational iteration algorithm-I with an auxiliary parameter (VIA-I with AP) [4,11], which is extremely basic, less demanding to execute and is likewise able to approximate the solution with high exactness and accuracy in a vast solution domain. For the convergence of this method, see [9].…”