Difference equation vs differential equation on different scales

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.

Variational principle for non-linear fractional wave equation in a fractal space

“…We will adopt the two-scale method [21][22][23][24] and the exp-function [25][26][27][28][29][30][31] to solve the fractional nonlinear wave equation. The two-scale transform [21][22][23][24] is an excellent tool to dealing with fractional differential equations in fractal space.…”

“…The two-scale transform [21][22][23][24] is an excellent tool to dealing with fractional differential equations in fractal space. The exp-function method [25][26][27][28][29][30][31] was proposed by Ji-Huan He, it is a powerful tool to finding approximate solutions of fractional differential equations. When…”

“…This paper is to establish a variational formulation for Eq. (1.1) using the semi-inverse method [1][2][3][4][5][6][7][8][9][10], and the two-scale transform [21][22][23][24] and the exp-function method [25][26][27][28][29][30][31] are used to find the approximate analytical solution of Eq. (1.1).…”

“…According to the definition of He's fractal derivative, eq. (3We adopt He's exp-function method[25][26][27][28][29][30][31] to solve eq. (3where  and  are real numbers.…”

Exact solutions of space-time fractional (2+1)-dimensional breaking soliton equation

“…In the past decades, many methods have been developed to obtain exact solutions of fractional-order NPDEs. Some of the most important methods are the homotopy perturbation method [1][2][3], the variational iteration method [4][5][6][7][8][9], and the exp-function method [10][11][12][13].…”