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Zbl 1234.35219
El-Wakil, S.A.; Abulwafa, E.M.; Zahran, M.A.; Mahmoud, A.A.
Time-fractional KdV equation: Formulation and solution using variational methods. (English)
[J] Nonlinear Dyn. 65, No. 1-2, 55-63 (2011). ISSN 0924-090X; ISSN 1573-269X/e

Summary: The semi-inverse method has been used to derive the Lagrangian of the Korteweg-de Vries (KdV) equation. Then the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann-Liouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to the Euler-Lagrange equation. Via Agrawal's method, one can easily derive the time-fractional KdV equation from this Euler-Lagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He's variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.

MSC 2000:: *35Q53 KdV-like equations
35R11
35C08
35A15 Variational methods (PDE)

Keywords: Riemann-Liouville fractional differential operator; Euler-Lagrange equation; Riesz fractional derivative; fractional KdV equation; He's variational-iteration method; solitary wave

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Zentralblatt MATH Berlin [Germany]