Difference equation vs differential equation on different scales
Ji‐Huan He, Fei-Yu Ji, Hamid M. Sedighi
Abstract
Purpose The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact. Design/methodology/approach A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales. Findings Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found. Originality/value The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.