Fourier Splitting Method for Kawahara Type Equations

Abstract

In this work, we integrate numerically the Kawahara and generalized Kawahara equation by using an algorithm based on Strang’s splitting method. The linear part is solved using the Fourier transform and the nonlinear part is solved with the aid of the exponential operator method. To assess the accuracy of the solution, we compare known analytical solutions with the numerical solution. Further, we show that astincreases the conserved quantities remain constant.

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

“…here u 3x ∶= u xxx , u 5x ∶= u xxxxx and 𝛾, p, and q are rescaled nondimensional quantities, n is an integer, n = 1 corresponds to the magneto-acoustic wave, and n = 2 to the shallow water wave [1]. When 𝛾 = 6, n = 1, p = 1, and q = 0, Equation (1.1) corresponds to the Korteweg-de Vries (KdV) equation.…”

“…Also, a lot of researchers have studied numerical solutions of the modified Kawahara equation. For instance, Fourier splitting method is used by Suarez and Morales [1], the meshless method of lines is proposed in Bibi et al [19], Gong et al [20] utilized multisymplectic Fourier pseudo-spectral method, collocation methods based on RBFs are developed by Zarebnia and Aghili [21] and Dereli and Dag [22], Bagherzadeh [23] applied B-spline collocation method, Crank-Nicolson differential quadrature method is applied by Korkmaz and Dag [24], Yuan et al [25] employed dual Petrov-Galerkin method, new multisymplectic integrator is introduced by Wen-Jun and Yu-Shun [26], and Marinov and Marinova [27] devised finite difference method [28]. Recently, RBF-finite difference method is used to solve Kawahara equation by Rasoulizadeh and Rashidinia [29], and differential quadrature method is applied to the Kawahara-type equations by Başhan [30].…”

“…In current paper, we shall focus on the following generalized type of the Kawahara equation: $$$$ \left\{\begin{array}{ll}{u}_t+\gamma {u}^n{u}_x+p{u}_{3x}-q{u}_{5x}& =0,\\ {}u\left(x,0\right)& =f(x),x\in \mathrm{\mathbb{R}},t\in \left[0,T\right],\end{array}\right. $$$$ here $$$ {u}_{3x}:= {u}_{xxx} $$$, $$$ {u}_{5x}:= {u}_{xxxxx} $$$ and $$$ \gamma, p $$$, and $$$ q $$$ are rescaled nondimensional quantities, $$$ n $$$ is an integer, $$$ n=1 $$$ corresponds to the magneto‐acoustic wave, and $$$ n=2 $$$ to the shallow water wave [1]. When $$$ \gamma =6 $$$, $$$ n=1 $$$…”