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    Numerical behaviour and solution of travelling wave and superposition of two solitary waves of the nonlinear KdV-Kawahara equation
    (Springer Basel Ag, 2025) Bashan, Ali
    In the present manuscript, the fifth-order nonlinear Korteweg-de Vries-Kawahara (KdV-Kawahara) equation including two dispersion terms is going to be handled. For the solution process, both finite difference and quadrature differential methods are used. The newly obtained results are important due to two aspects. First, they are better than those of most available ones. The second and most important, the results are found at the low cost of computational efforts. These important aspects ensure that they can also be applied to a wide range of problems encountered in various fields of applied sciences. Moreover, the computed results get closer to the exact ones when the step sizes refine. Travelling single solitary wave and the superposition of two solitary waves solutions are obtained numerically. A comparison of the numerical results shows that present algorithm by contribution of the two numerical methods obtained higher accurate solutions successfully.
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    Numerical features of the fifth-order nonlinear Kawahara equation for modeling shallow water waves
    (Aip Publishing, 2025) Bashan, Ali
    In the present study, numerical solutions of the important type of the fifth-order Korteweg-de Vries (KdV) equation for modeling shallow water waves, namely, Kawahara equation, are investigated. With the solutions of the higher order Kawahara equation used in the modeling of shallow water waves, the behavior change of the wave with parameter changes is revealed. Unlike the classical numerical methods, two effective numerical methods are combined and used together. One of the components of the scheme is differential quadrature method (DQM) that approaches the derivative of the mesh points in the solution domain. In other words, space discretization is completed by using DQM. The other component is the Crank-Nicolson scheme, which is the effective type of the finite difference method. By using the advantages of the DQM and Crank-Nicolson scheme with effective linearization technique together, high accurate solutions are obtained. Four different test problems and their various forms are solved numerically. Error norms, central processing unit times, rate of convergence, and three invariants are calculated and reported. Published under an exclusive license by AIP Publishing.https://doi.org/10.1063/5.0276849I