New Analytical Approximations with Error Estimates for the One-loop Soliton Solution to the Vakhnenko Equation

Abstract

In this paper, based on the traditional homotopy analysis method, we have successfully proposed a new analytical method, namely the piecewise perturbation method (PPM), for solving nonlinear problems. Here we introduce the idea of Newton iteration into the traditional homotopy analysis method to revise the initial guesses, which significantly increases the accuracy and the convergence rate of the series solutions. Further, we apply this method to obtain the one-loop soliton solution of the Vakhnenko equation in order to verify its potential and validity in solving nonlinear problems. With the aid of the optimal value of the convergence-control parameter determined by the averaged residual error technique, comparisons are made between the proposed method and the traditional homotopy analysis method. The results reveal that these new approximations with error estimates possess better accuracy and higher convergence rate than those obtained by the traditional homotopy analysis method.