Numerical solution of the Gardner equation using quartic B-spline collocation method

In this paper, we will consider one numerical solution to solve the nonlinear Gardner equation. The quartic B-spline (QBS) collocation method will be used to determine the unknown term in this equation. In this regard, we apply the quasilinearization technique to linearize the nonlinear terms of the equation and then, combine the QBS collocation method in space with the finite difference in time. This operation provides an efficient explicit solution with high accuracy and minimal computational effort for this problem. It is further proved that the proposed method has the order of convergence $\textit{O}(k+h^2)$. Also, the method is shown to be unconditionally stable using the Von-Neumann method. Finally, the efficiency and robustness of the proposed approach for solving the nonlinear Gardner equation are demonstrated by one numerical example. These numerical computations will be compared to radial basis functions (RBFs).