Purpose -The purpose of this paper is to obtain soliton solution of the Kaup-Kupershmidt (KK) equation with initial condition. The most important feature of this method is to obtain the solution without direct transformation. Design/methodology/approach -In this paper, the homotopy perturbation method (HPM) is used for obtaining soliton solution of the KK equation. The numerical solutions are compared with the known analytical solutions. The results of numerical examples are presented and only a few terms are required to obtain accurate solutions. Results derived from this method are shown graphically.Findings -The authors obtained the one soliton solution for the KK equation by HPM. The numerical results showed that this method is very accurate. The HPM provides a reliable technique that requires less work if compared with the traditional techniques and the method does not also require unjustified assumptions, linearization, discretization or perturbation. The HPM is very easily applied to both differential equations and linear or nonlinear differential systems. Originality/value -The paper describes how the authors obtained one soliton solution for the KK equation by HPM. The numerical results presented show that this method is very accurate.

In this paper, we studied the solitary wave solutions of the (2+1)-dimensional Boussinesq equation utt −uxx−uyy−(u2)xx−uxxxx = 0 and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation uxt −6ux 2 +6uuxx −uxxxx −uyy −uzz = 0. By using this method, an explicit numerical solution is calculated in the form of a convergent power series with easily computable components. To illustrate the application of this method numerical results are derived by using the calculated components of the homotopy perturbation series. The numerical solutions are compared with the known analytical solutions. Results derived from our method are shown graphically.

In this paper, we consider the nonlinear vibrations of nano-sized cantilever. The elastic force is considered anharmonic, deriving from a Morse potential and the nonlinearity is attributed to the Casimir force. The solution is established for viscous and fractional damping by making use of He's polynomials which are calculated from homotopy perturbation method (HPM). The solution procedure explicitly reveal the complete reliability and simplicity of the proposed algorithm. Moreover, comparison with variational iteration method (VIM) shows that both the techniques are in full agreement with each other.