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.
Highlights
The study of nonlinear phenomena has always been an active subject in applied science and physics
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 Korteweg-de Vires (KdV) equation
Noting that most of above work is analytical/semianalytical, 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
Summary
The study of nonlinear phenomena has always been an active subject in applied science and physics. 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 θ-weight finite difference method was used to approximate the first-order time derivative. 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 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. Noting that most of above work is analytical/semianalytical, 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. Some nonlinear scale functions are considered, which makes the KdV equation more general
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