Finite difference/collocation method for a generalized time-fractional KdV equation

Abstract

In this paper, we studied the numerical solution of a time-fractional Korteweg-de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi-Gauss-Lobatto (JGL) nodes was applied to solve this equation and the corresponding stability was analyzed theoretically, while the convergence was verified numerically. Furthermore, we investigated the behavior of solution of the generalized KdV equation depending on its parameter δ, scale function z(t) in fractional derivative. We found that the full discrete scheme was effective to obtain a numerical solution of the new KdV equation with different conditions. The wave number δ in front of the third order space derivative term played a significant role in splitting a soliton wave into multiple small pieces.