Finite Difference/Collocation Method for a Generalized Time-Fractional KdV Equation

Abstract

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.

High-order approximation to generalized Caputo derivatives and generalized fractional advection-diffusion equations

“…(2.15) Lemma 2.2. If the scale function ζ fulfills (2.4), and the weight function ω is non-negative and non-decreasing on uniform time grids, then (i) [26] Linear approximation coefficient satisfies,…”

“…In [18], authors presented numerical scheme for the generalized fractional telegraph equation in time. Cao et al [26] worked on the generalized time-fractional Kdv equation. Xu et al [27] considered the solution of generalized fractional diffusion equation.…”

Numerical method for generalized time fractional KdV‐type equation

“…For truncation error (3.3), in order to overcome the difficult caused by nonlinearity of scale function, Newton's interpolation method can be utilized to generate a satisfying approximation. For instance, see for more details of linearization.Lemma () . Denote by U ( ξ ) = σ ( z −1 ( ξ )) u ( x , z −1 ( ξ )) for ξ ∈ [ z 0 , z M ].…”

“…A finite difference scheme with the same accuracy as L 1‐scheme for the approximation of generalized derivative is proposed. In , a generalized time‐fractional KdV equation using finite difference/collocation method is studied. In , numerical solutions for one‐dimensional time‐fractional diffusion and Burgers equations with generalized fractional derivative are derived by finite difference method.…”

“…In , a homotopy perturbation method is implemented for solving the fractional KdV equation. Other than semi‐analytical methods, a finite difference/collocation method is constructed for studying generalized time‐fractional KdV equation in . In , a linearized time‐fractional KdV equation on unbounded spatial domains is studied with an exact artificial boundary conditions which reduces the problem into an equivalent problem on a finite interval.…”