Numerical solutions of the Gardner equation by extended form of the cubic B-splines

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. The optimum value of the extension parameter can be determined by scanning some intervals containing zero. This study aims to solve some initial boundary value problems constructed 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 whether the test problems remain constant as expected from the theoretical aspect. The stability of the proposed method is investigated by the von Neumann analysis.