Study of the Gardner equation with homogeneous boundary conditions via fourth order modified cubic B-spline collocation method

In this work, the well-known Gardner equation is converted into a coupled system of nonlinear partial differential equations and a modified cubic B-spline collocation method has been applied to find its numerical solution. Time discretization and linearization of Gardner equation has been carried out using Crank–Nicolson method and quasi-linearization respectively. A linear system of algebraic equations is obtained which is found to be unconditionally stable by Von Neumann analysis. Numerical investigations are performed on the Gardner equation subjected to homogeous boundary conditions in different situations such as propagation of initial positive pulse and kink like wave, propagation and interaction of two solitons, wave production from a single soliton, evolution of non-propagating solitons. The results obtained are compared with those available in the literature and are found to be better. The conserved quantities are also computed to show that the conservation laws are preserved as expected from the theoretical aspect. Numerical results demonstrate the accuracy and validity of the present method which can be further applied to solve other nonlinear problems.