# Exponential B-spline Collocation Solutions to the Gardner Equation

## TL;DR

This paper introduces a collocation method using exponential B-splines to solve the Gardner equation, effectively handling its nonlinear terms and coupled PDE system with verified accuracy and conservation properties.

## Contribution

The paper develops a novel exponential B-spline collocation approach for the Gardner equation, including order reduction and integration with Crank-Nicolson, demonstrating its effectiveness.

## Key findings

- High accuracy in solving the Gardner equation
- Effective handling of nonlinear quadratic and cubic terms
- Preservation of conservation laws during simulations

## Abstract

The exponential B-spline basis function set is used to develop a collocation method for some initial boundary value problems (IBVPs) to the Gardner equation. The Gardner equation has two nonlinear terms, namely quadratic and cubic ones. The order reduction of the equation is resulted in a coupled system of PDEs that enables the exponential B-splines to be implemented. The system is integrated in time by Crank-Nicolson implicit method. The validity of the method is investigated by calculating the discrete maximum error norm and observing the absolute relative changes of the conservation laws at the end of the simulations.