Exponential B-spline collocation solutions to the Gardner equation

Exponential B-splines are used to set up a collocation method for solving the Gardner equation. The space reduction of the Gardner equation is carried out to be able to obtain an exponential B-spline approximation for the collocation method. Thus, a coupled system is integrated using the Crank-Nicolson implicit method in time together with the first-order linearization method and then the collocation method is applied to have a linear algebraic system. This system is shown to be stable by using the Von Neumann analysis. The discrete maximum errors are found fairly small and relative changes of the conservation laws remain constant during simulations for the text problems.