Abstract
The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg‐de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameter ħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th‐order and 14th‐order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement.
Highlights
It is difficult to solve nonlinear problems, especially by analytic technique
The homotopy analysis method Homotopy Analysis Method (HAM) 1, 2 is an analytic technique for nonlinear problems, which was first introduced by Liao in 1992
This method has been successfully applied to many nonlinear problems in engineering and science, such as the magnetohydrodynamics flows of nonNewtonian fluids over a stretching sheet 3, boundary layer flows over an impermeable stretched plate 4, nonlinear model of combined convective and radiative cooling of a spherical body 5, exponentially decaying boundary layers 6, and unsteady boundary Journal of Applied Mathematics layer flows over a stretching flat plate 7
Summary
The homotopy analysis method HAM 1, 2 is an analytic technique for nonlinear problems, which was first introduced by Liao in 1992. The Burgers equation is a fundamental partial differential equation from fluid mechanics It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow. We employ the homotopy analysis method to obtain the solutions of the Korteweg-de Vries KdV and Burgers equations so as to provide us a new analytic approach for nonlinear problems. Let u0 r, t denote an initial guess of the exact solution u r, t , ħ / 0 an auxiliary parameter H r, t / 0 an auxiliary function, and an auxiliary linear operator, Q ∈ 0, 1 as an embedding parameter by means of homotopy analysis method, we construct the so-called zeroth-order deformation equation. Note that homotopy analysis method contains the auxiliary parameter ħ, which provide us with that control and adjustment of the convergence of the series solution 2.6
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
