Abstract
Tsunamis are strong waves, arisen from volcanic eruptions, landslides, or earthquakes sweeping across oceans. The geophysical Korteweg-de Vries (gKdV) equation which governs the tsunami wave propagation in oceans is investigated in this work using an improved exp(-F(η))-expansion method. Shooting and adaptive moving approaches are taken into account. We retrieve several new solitary solutions for the gKdV equation. The obtained solution from implementing shooting method is successfully used as an initial value for the adaptive approach which is utilized to construct the numerical solution of the proposed problem. The constructed exact solutions coincide with the obtained numerical solutions. The accuracy of the presented numerical approximations is discussed. We apply Fourier concept to explore the accuracy and stability of the numerical schemes which is unconditionally stable. A clear comparison between the analytic and numerical outcomes is presented via some 2D and 3D sketches which are depicted under special selections of some parameters. Moreover, we illustrate the relative error and CPU time for the numerical technique. The proposed approaches can be easily utilized to deal with other partial differential equations.