# Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation
## 1 Introduction  
The study of nonlinear phenomena has always been an active subject in applied science and physics. As a classic nonlinear partial differential equation, the Korteweg-de Vires (KdV) equation is very important in analyzing waves on shallow water surfaces. In history, the KdV equation was first introduced by [PERSON] around 1877, and rediscovered by [PERSON] and [PERSON] in 1895 [1]. The KdV equation was firstly derived as an evolution equation that governs small-amplitude, long-surface gravity waves propagating in a shallow channel of water. However, the KdV equation can only be solved analytically in selected initial and boundary conditions due to its nonlinearity [2; 3; 4; 5]. With the development of computational techniques, the KdV equation gained much attention of mathematicians and physicists in the past few years. In [6], by choosing appropriate base functions, a Legendre-Petrov-Galerkin method is applied for the KdV equation. An optimal rate of convergence in \(L^{2}\)-norm is derived without the nonlinear part. By using different test and trial function spaces and basis functions, [7] proposes a new dual-Petrov-Galerkin method for high odd-order differential equations. In [8], pseudo-spectral method is applied to solve the KdV equation, in which the spectral method is implemented parallelled with split-step. This is one of the original results of the pseudo-spectral method applied to the KdV equation. Later in [9], the KdV equation is solved by the homotopy perturbation method. By constructing special forms of initial conditions, numerical solutions expressed by the trigonometric functions, hyperbolic functions are obtained. Many authors applied the collocation method to solve KdV equation due to its high accuracy and good performance in the nonlinear case. In [10], the KdV equation is solved by using the radial basis functions. It is a meshless method based on the collocation with radial basis functions. Five standard radial basis functions are used in the method of the collocation. In [11], A KdV-Burgers equation wasdiscussed by using the modified Bernstein polynomials. The B-polynomials were used to expand the desired solution requiring discretization with only the time variable. The Galerkin method was used to determine the expansion coefficients to construct initial trial functions. A fourth-order Runge-Kutta method was used to solve the system of equations for the time variable. In [12], the sinc-collocation method was applied to solve the nonlinear third-order KdV equation in the space direction with periodic forcing at the boundary numerically and a \(\theta\)-weight finite difference method was used to approximate the first-order time derivative. However, all the work above is on the KdV equation with integer-order derivatives. In the recent forty years, many partial differential equations are reconsidered by changing their classical derivative into fractional derivatives. Fractional derivative is not a new concept. In fact, it was firstly proposed in the discussion of [PERSON] and [PERSON] in 1695. Since 1974, when the first international conference of fractional calculus and its application was held, many researchers from scientific and engineering fields brought fractional derivatives back to their research interests. A large number of physical models were renewed by fractional calculus. Under such a background, fractional KdV equation, as well as many other fractional PDEs, are also studied by different methods, either analytical or numerical techniques [13; 14; 15; 16]. In [17], a fractional KdV equation with a Caputo derivative is solved by the Adomian decomposition method. The approximation solution is presented via a convergent series with easily computable components. In [18], based on Maple, the Adomian decomposition method is extended to derive explicit and numerical solutions of the fractional KdV-Burgers equation.