# Numerical Solutions of the Gardner Equation by Extended Form of the Cubic B-splines

## TL;DR

This paper introduces an extended cubic B-spline collocation method to numerically solve the Gardner equation, aiming to improve accuracy and efficiency over classical methods, validated through analytical and stability analyses.

## Contribution

The study develops an extended cubic B-spline approach for the Gardner equation, optimizing the extension parameter to enhance solution accuracy and stability.

## Key findings

- The method accurately solves initial boundary value problems for the Gardner equation.
- Conservation laws are maintained throughout the simulations.
- Von Neumann analysis confirms the method's stability.

## Abstract

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. Determination of the optimum value of the extension parameter can be achieved by scanning some intervals containing zero. This study aims to solve some initial boundary value problems con- structed for the Gardner equation by the extended cubic B-spline collocation method.The test problems are derived from some analytical studies to validate the efficiency and accuracy of the suggested method. The conservation laws are also determined to observe them remain constant as expected in theoretical aspect. The stability of the proposed method is investigated by the Von Neumann analysis.