Numerical Solutions of the Gardner Equation by Extended Form of the Cubic B-splines
This is an incremental improvement for researchers using numerical methods to solve nonlinear wave equations.
The authors apply an extended cubic B-spline collocation method to solve the Gardner equation, achieving improved accuracy over classical B-splines by optimizing an extension parameter. The method is validated on test problems and demonstrates stable conservation of invariants.
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.