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

Abstract

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.

Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

“…The rest of the paper is structured as follows: In Section 2, based on the basic system of equations of ion-acoustic solitary waves, we obtain a new 3D Schamel-KdV equation by using multi-scale analysis and the perturbation method [42]. A new 3D TSFSchamel-KdV equation is obtained in Section 3 according to the new integer-order model and by using the semiinverse method and the fractional variational principle [43,44]. In Section 4, applying the Riemann-Liouville fractional derivative [39,40], we discuss the conservation laws of the new fractional model.…”

“…Definition 2 (see [44]). The left Riemann-Liouville fractional derivation of a function ( , , , ) is defined as…”

Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions

“…Then, Agrawal's method (Agrawal 2002;Muslih and Agrawal 2010) is used by El-Wakil et al (2010) to derive the TFKdV equation in the form (see Appendix)…”

Formulation and Solution of Time-Fractional Degasperis-Procesi Equation via Variational Methods

“…As an abstract bi-Hamiltonian evolution equations with infinitely many conservation laws, the Degasperis-Procesi equation has obtained by Johnson [6], Dullin et al [7] has proved to be an approximate model of shallow water wave propagation in the small amplitude and long wavelength regime, Fokas and Fuchssteiner [8], Lenells [9], Camassa and Holm [10] put it forward the derivation of solution as a model for dispersive shallow water waves and discovered that it is formally integrable dimensional Hamiltonian system and that its solitary waves are solitons. Most of classical mechanics techniques have studied conservative systems, but almost of the processes observed in the physical real world are nonconservative [11]. During the past three decades or so, fractional calculus has obtained considerable popularity and importance as generalizations of integer-order evolution equations, and used to model problems in neurons, hydrology, viscoelasticity and rheology, image processing, mechanics, mechatronics, physics, finance and control theory, see [12,13,14,15,16,17,18,19].…”