Reliable iterative methods for 1D Swift–Hohenberg equation

Abstract

In this paper, the nonlinear problem of the 1 D Swift-Hohenberg equation (S-HE) has been solved by using five reliable iterative methods. The first one is the Daftardar-Jafari method namely (DJM), the second method is the Temimi-Ansari method namely (TAM), the third method is the Banach contraction method namely (BCM), the fourth method is the Adomian decomposition method namely (ADM) and finally the fifth method is the Variational iteration method (VIM) to obtain the approximate solutions. In this work, we discussed and applied these iterative methods to solve the S-HE and compared them. In addition, the fixed-point theorem was given to illustrate the convergence of the five methods. To illustrate the accuracy and efficiency of the five methods, the maximum error remainder was calculated since the exact solution is unknown. The results showed that the five iterative methods are accurate, reliable, time saver and effective. All the iterative processes in this paper implemented in MATHEMATICA®11.

Keywords

• approximate solution
• iterative methods
• nonlinear partial differential equations
• semi-analytical solution
• swift-hohenberg equation