THE VARIATIONAL ITERATION METHOD WHICH SHOULD BE FOLLOWED

Abstract

This paper proposes three standard variational iteration algorithms for solving differential equations, integro-differential equations, fractional differential equations, fractal differential equations, differential-difference equations and fractional/fractal differential-difference equations. The physical interpretations of the fractional calculus and the fractal derivative are given and an application to discrete lattice equations is discussed. The paper then examines the acceleration of some iteration formulae with particular emphasis being placed on the exponential Padé approximant that is suggested for solitary solutions and the sinusoidal Padé approximant that is usually used for periodic and compacton solutions. The paper points out that there may not be any physical meaning to the exact solutions of many nonlinear equations and stresses the importance of searching for approximate solutions that satisfy both the equations and the appropriate initial/boundary conditions. The variational iteration method is particularly suitable for solving this kind of problems. Approximate initial/boundary conditions and point boundary initial/conditions are also discussed, with the variational iteration method being capable of recovering the correct initial/boundary conditions and finding the solutions simultaneously.