Time-fractional KdV equation: formulation and solution using variational methods

• Published in: Nonlinear dynamics Vol. 65; no. 1-2; pp. 55 - 63
• Main Authors: El-Wakil, S. A., Abulwafa, E. M., Zahran, M. A., Mahmoud, A. A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-07-2011; Springer Nature B.V
• Subjects: Automotive Engineering; Classical Mechanics; Control; Derivatives; Differential equations; Dispersions; Dynamical Systems; Engineering; Euler-Lagrange equation; Initial conditions; Inverse method; Korteweg-Devries equation; Mathematical analysis; Mechanical Engineering; Nonlinearity; Operators; Operators (mathematics); Original Paper; Solitary waves; Variational methods; Vibration; Solitary wave; Riemann–Liouville fractional differential operator; Euler–Lagrange equation; Fractional KdV equation; He’s variational-iteration method; Riesz fractional derivative
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-010-9873-5

In this work, 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 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.