Analytical solution of time fractional Kawahara and modified Kawahara equations by homotopy analysis method. (English) Zbl 1499.65605
Summary: The article covers the implementation of the Homotopy Analysis method (HAM) in order to derive the general estimated analytical solutions for the non-linear time-fractional Kawahara and modified Kawahara equations. The Kawahara equations play a salient role in describing the formation of non-linear water waves in the long wavelength region.To illustrate the applicability and effectiveness of derivative with fractional order to portray the water waves in long wavelength regime,we have presented numerical outcomes graphically. The obtained observations have further fortified the idea that the suggested method offers a powerful and easy way to solve such non-linear system of equations. Besides, its main advantage is that it offers a series solution without any discretization or linearization. The numerical analysis illustrates that the surfaces obtained from the HAM technique are closely resemble the exact solution of the considered equations. Moreover, we have analyzed our solution based on the degree enhancement of time intervals so as to observe the dynamics of our solution with the respective time variations. The numerical solutions converge to the exact solution of the Kawahara equations using convergence analysis, validating the significance of our suggested method.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35R11 | Fractional partial differential equations |
Keywords:
homotopy analysis method (HAM); formation of water waves; Riemann-Liouville operator; Caputo derivative; convergence of HAM; time fractional Kawahara and modified Kawahara equationsReferences:
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