Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries

• Group classification of variable-coefficient KdV equations is performed.
• Classification of similarity solutions is presented.
• Lie symmetries are used to reduce a BVP for KdV equations to one for ODE.
• The BVP for ODE is solved numerically using the finite difference method.
• Numerical solutions are computed and the vast parameter space is studied.

The exhaustive group classification of a class of variable coefficient generalized KdV equations is presented, which completes and enhances results existing in the literature. Lie symmetries are used for solving an initial and boundary value problem for certain subclasses of the above class. Namely, the found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations (which are PDEs) to an initial value problem for nonlinear third-order ODEs. The latter problem is solved numerically using the finite difference method. Numerical solutions are computed and the vast parameter space is studied.