Soliton solutions for the fifth-order KdV equation with the homotopy analysis method

Abstract

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the fifth-order KdV equation. The homotopy analysis method (HAM) provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of series solution.

The homotopy analysis method and the Liénard equation

“…Liao [15,16] has developed this purely analytic technique to solve nonlinear problems in science and engineering. The HAM has been applied successfully to many nonlinear problems such as free oscillations of self-excited systems [17], the generalized Hirota-Satsuma coupled KdV equation [18], heat radiation [19], finding the root of nonlinear equations [20], finding solitary-wave solutions for the fifth-order KdV equation [21], finding solitary wave solutions for the Kuramoto-Sivashinsky equation [22], finding the solitary solutions for the Fitzhugh-Nagumo equation [23], boundary-layer flows over an impermeable stretched plate [24], unsteady boundarylayer flows over a stretching flat plate [25], exponentially decaying boundary layers [26], a nonlinear model of combined convective and radiative cooling of a spherical body [27], and many other problems (see [28,29,30,31,32,33,34,35,36], for example).…”

Unsteady mixed convection flow of nanofluid on a rotating cone with magnetic field

“…The viscous flow equations for unsteady flow are presented along with heat and mass transfer analysis. The governing highly nonlinear coupled partial differential equations are first transformed to coupled ordinary differential equations using the similarity transformations and then solved analytically with the help of homotopy analysis method (Liao , 2004(Liao , 2005(Liao , 2009Liao and Cheung 2003;Abbasbandy 2006Abbasbandy , 2008Abbasbandy and Samadian 2008;Ellahi and Riaz 2010;Ellahi and Afzal 2009;Nadeem et al 2010;Nadeem and Hussain 2009;Nadeem and Saleem 2013). The influences of different physical parameters are also presented and discussed graphically.…”

Series Solution of the System of Integro-Differential Equations

“…This technique has been successfully applied to many nonlinear problems such as the viscous flows of non-Newtonian fluids [3,4], the Korteweg-de Vries-type equations [5,6], nonlinear heat transfer [7,8], finance problems [9,10], Riemann problems related to nonlinear shallow water equations [11], projectile motion [12], Glauertjet flow [13], nonlinear water waves [14], groundwater flows [15], Burgers-Huxley equation [16], time-dependent Emden-Fowler type equations [17], differential-difference equation [18], Laplace equation with Dirichlet and Neumann boundary conditions [19], thermal-hydraulic networks [20], and recently for the Fitzhugh-Nagumo equation [21], and so on. On the other hand, one of the interesting topics among researchers is solving integro-differential equations.…”