时间分数阶KdV方程的变分形式与求解
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
摘要：
在这项工作中，半逆方法已被用来推导的拉格朗日的Korteweg-de Vries（KdV）方程。然后，将KdV方程的拉格朗日量的时间算子转化为分数阶左Riemann-Liouville分数阶微分算子。该拉格朗日量的泛函的变分巧妙地导致欧拉-拉格朗日方程。通过Agrawal的方法，可以很容易地从Euler-Lagrange方程导出时间分数阶KdV方程。值得注意的是，在所得到的KdV方程中的时间分数项在Riesz分数阶导数中以直接的方式获得。作为第二步，导出的时间分数KdV方程求解使用他的变分迭代法。计算采用的初始条件取决于KdV方程的非线性系数和色散系数。我们注意到，更显着的效果和更深入的了解所产生的孤波的形成和性质，另外考虑分数阶导数旁边的非线性和色散项。
英文摘要：
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.
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