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

TL;DR

This paper classifies variable coefficient generalized KdV equations using Lie symmetries and employs numerical methods to solve the resulting boundary value problems, providing new insights into their solutions.

Contribution

It completes the group classification of these equations and applies Lie symmetries to reduce and numerically solve boundary value problems.

Findings

Numerical solutions for the reduced ODEs are obtained.

The parameter space of solutions is extensively studied.

Lie symmetries effectively reduce complex PDEs to solvable ODEs.

Abstract

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.