Two effective methods for solution of the Gardner–Kawahara equation arising in wave propagation

In this paper, we deal with the analytical and numerical solutions of the Gardner–Kawahara (G-K) equation, known as the extended Korteweg–de Vries equation, which describes solitary-wave propagation in media and occurs in the notion in plasmas and in notion of shallow water waves with surface tension and notion of magneto-acoustic waves. This study is split into two primary parts: in the first part, (G′/G) and exp (−ϕ(ξ)) expansion methods are employed to obtain some novel exact solutions of the G-K equation. Furthermore, some hypotheses are used to get different exact traveling wave solutions (including Hyperbolic,Trigonometric and Rational solutions). (G′/G)-expansion method is a comprehensible and noncomplex technique used to procure traveling wave solutions of partial differential equations (PDEs). By using the (G′/G)-expansion technique, solution of PDEs can be indicated as a series and the coefficients of the series can be procured by solving a set of algebraic equations. This method is highly effective, requiring only the solution of a few algebraic equations to create the traveling wave solution (Ali and Tag-eldin, 2023 [3]). The exp (−ϕ(ξ))- expansion method is used as the first time to examine the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator–prey system. The method also has been used for many other nonlinear evolution equations (Abdelrahman et al., 2015 [1]). In the second part, finite element method is dedicated to deriving numerical solutions of the equation. In the same part, based on septic B-spline approximation, a collocation method has been offered and applied for numerical solutions of G-K equation conceiving different parameter values of test problem. To discretize time and space derivatives, forward finite difference and Crank–Nicolson approximations have been used, respectively. Unconditional stability is proved by using Von-Neumann method. B-spline functions have some reliable properties like smoothness, local support and ability of handling local phenomena, which make them proper to solve linear and nonlinear partial differential equations easily and precisely. Collocation technique has two great superiorities: constituting method does not involve integrations and the resulting matrix system is banded with small band width. For this reason, B-splines when associate with the collocation technique ensure a simple solution process of linear and nonlinear PDEs. An example is successfully solved by calculating the error norms L2 and L∞ for illustrating the proficiency and reliability of the method and highlighted the significance of this work. Also to reflect the efficiency of this method for solving the nonlinear equation, the results are depicted both graphically and in tabular form. Calculated results are found to be in good agreement with the exact solutions and other techniques given in the literature.