Petrov–Galerkin method and K(2, 2) equation

Abstract

In this paper a Petrov–Galerkin method is used to derive a numerical solution for the K(2, 2) equation, where we have chosen cubic B-splines as test functions and piecewise linear functions as trial functions. The product approximation technique is applied to the nonlinear terms. A Crank–Nicolson scheme is used to discertize in time. A nonlinear pentadiagonal system is obtained. We solve this system by Newton’s method and a linearization technique. The accuracy and stability of the scheme have been investigated. The single compacton solution and the conserved quantities are used to assess the accuracy of the scheme. The interaction of two compactons are displayed and the numerical results have shown that this compacton exhibits a soliton-like solution.

Keywords:

• Compacton
• Petrov–Galerkin method
• Linearization