Numerical algorithms for solutions of Korteweg–de Vries equation

Abstract

The nonlinear Korteweg-de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion-based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples.

Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method

“…Ali Bashan [32] and Korkmaz [33] used differential quadrature method for numerical study of modified Burgers' equation and transport of conservative pollutants respectively. Some more work on DQM can be seen in [34][35][36] .…”

Geometric numerical schemes for the KdV equation

“…The numerical discretization of the KdV equation can be done by finite differences [63,58,33], finite volumes [6,22], finite elements [3,9], discontinuous Galerkin [39] and, of course, by spectral methods [43,29,65,36]. However, recently the so-called geometrical numerical discretizations have been developed [46,28,40,38,34,35,16].…”

An effective application of differential quadrature method based onmodified cubic B-splines to numerical solutions of the KdV equation

“…The DQM has widely become a preferable method in recent years due to its simplicity for application. Numerous researchers have developed different types of DQMs by utilizing various test functions such as Legendre polynomials and * Correspondence: alibashan@gmail.com 2010 AMS Mathematics Subject Classification: 65D07, 65L20, 65M99, 65L06 spline functions [9,10], Lagrange interpolation polynomials [23,29,30], Hermite polynomials [14], radial basis functions [32], harmonic functions [37], Sinc functions [12,24], and B-spline functions [5-8, 20, 25, 26].…”