A NOVEL PERSPECTIVE FOR SOLUTIONS OF THE GARDNER EQUATION USING TWO RELIABLE METHODS

The main goal of this article is to find different and new analytical and numerical solutions of the Gardner equation. The equation, also known as the combined KdV–mKdV equation, clarifies several interesting physical phenomena, such as internal waves in a stratified ocean [1], long wave propagation in an inhomogeneous two-layer shallow liquid [2], and ion acoustic waves in plasma with negative ions [3]. Our study consists of two main parts: in the first part, (G'/G)- expansion method is implemented to create some alternative exact solutions of the Gardner equation. Also in this part, various hypotheses are considered to construct many exact traveling wave solutions (involving Hyperbolic, Trigonometric, and Rational solutions). In the second part, a collocation method based on the septic B-spline approximation has been introduced and put into practice for the numerical solution of the equation while taking various test problem parameter values into account. Additionally, Von-Neumann stability analysis has been carried out, ensuring the scheme’s unconditional stability. The appropriate solutions for the two test problems are found by computing the L_2 and L_∞ error norms, which highlight the significance of the procedure and demonstrate its applicability and credibility. The numerical findings are inferred to match the analytical answers well, suggesting that the existing B-spline collocation algorithm is strong and appealing. The results are tabulated and reported both modally and in terms of the productivity of the procedure. The results produced from both analytical and numerical methods demonstrate the great utility of this study for scientists tasked with identifying characteristics and features of nonlinear processes across a variety of scientific domains.