  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.
