In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+\alpha}(\mathbb{R})), where (\alpha \in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent $\alpha$. Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.

翻译：本文提出并分析了用于求解涉及分数阶拉普拉斯算子的分数阶Korteweg-de Vries（KdV）方程初值问题的全离散有限差分格式。我们通过引入与连续算子相容且具备若干有助于收敛性分析的特定性质的离散分数阶拉普拉斯算子来设计该格式。在初始数据(u_0 \in H^{1+\alpha}(\mathbb{R}))（其中(\alpha \in [1,2))）的假设下，我们的研究证明了全离散有限差分格式获得的近似解收敛到分数阶KdV方程的经典解。通过针对不同分数阶指数$\alpha$的若干数值算例验证了理论结果。此外，我们还证明了Crank-Nicolson有限差分格式在保持内在守恒量的同时具有改进的收敛速度。