# Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation
## 1 Introduction
In [19], the homotopy analysis method that was developed for integer-order differential equations is directly extended to derive explicit and numerical solutions of the nonlinear fractional KdV equation. In [20], the homotopy perturbation method is implemented to construct solitary solutions for variants of the KdV equations with fractional time derivatives. In this scheme, the solution takes the form of a convergent series, and its terms can be easily calculated via iteration. In a recent work [21], the explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with time/space-fractional derivatives were presented and discussed. The numerical method is a technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Noting that most of above work is analytical/semiaalytical, we are interested in solving the fractional KdV equation numerically by a finite difference/collocation method, and trying to discover some particular features of the soliton solution of the fractional KdV equation. Fractional derivatives have various definitions, out of which the frequently used ones are defined in Riemann-Liouville and Caputo senses, which can be regarded as a convolution between a fractional power kernel and a function. These kinds of fractional power kernel put more weight in the present than in the past, which is particularly suitable to describe physical processes in the material with short memory, but is incapable of properly describing some behavior observed in materials with huge heterogeneities and structures with different scales. In [22], a new kind of fractional KdV model was established by using the developed Caputo-Fabrizio fractional derivative, and the conditions of existence and uniqueness of the continuous solution were provided. In [23], a time-fractional Shamel-KdV equation with Riesz fractional derivatives was derived to describe nonlinear behavior of dust-ion acoustic waves. In 2012, to unify different existing fractional derivatives, a new fractional calculus was proposed in [24]. Taking a scale function and a weight function into consideration, fractional integrals and derivatives were essentially generalized. Partial differential equations with such generalized fractional derivatives are not studied much [25; 26; 27; 28]. Therefore, in this paper, we choose the KdV equation to further study this topic. By using a generalized derivative with respect to scale function and weight function, we defined a new time-fractional KdV equation. As there is strong nonlinearity in the structure of the KdV equation, the collocation method is applied to the space direction while the finite difference method is implemented in the time direction. Because of the complexity of generalized fractional derivative, some classic mathematical tools are not available anymore in studying the generalized KdV equation, such as variational principle, energy functional, Holder estimation and Sobolev-type inequalities. We only have very limited methods of theoretical analysis to detect such KdV equations. However, in the current circumstance, the numerical method shall be a useful and direct way to show the details of the considered KdV equation with generalized fractional derivatives. After applying the collocation method in the space direction, the generalized fractional KdV equation is reduced to a nonlinear generalized fractional differential equation with respect to \(t\). Reference [29] studied the existence and uniqueness of a class of generalized fractional boundary value problems by using topological degree theory. In our paper, we focus on numerical methods to analyze such a KdV equation. Thus, we choose initial and boundary conditions that are sufficiently good to guarantee the existence and uniqueness of the new time-fractional KdV equation. We apply a semi-discretization scheme to solve this new fractional KdV. Based on numerical simulation, many new characteristics of the new fractional KdV equation are presented.