Abstract
Exact analytical form and approximate form of different kinds of tsunami wave solutions of the fractional geophysical Korteweg-de-Vries (fgKdV) equation are derived. The fgKdV equation is constructed by replacing the Caputo fractional derivative from the geophysical Korteweg-de-Vries equation. Some important definitions and results of the Caputo fractional derivative are discussed. The Darboux transformation and Residual power series method are used to derive some exact solutions and approximate analytical solutions to the fgKdV equation. Using the Darboux transformation the single solitary tsunami wave, singular periodic tsunami wave, two-soliton tsunami wave, singular solitary tsunami wave, solitary periodic tsunami wave, rational tsunami wave, rational solitary tsunami wave, and rational periodic tsunami wave were obtained. By performing the Residual power series method the approximate analytical solution is obtained. The graphical presentations of several exact solutions are obtained by Darboux transformation and Residual power series method. It is seen that the fractional parameter has a significant effect on the structure of the nonlinear tsunami waves of the fgKdV equation. The results of this study are applicable to understand the dynamics of multiple tsunami wave described by the fgKdV equation in the ocean.
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