Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

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 possesses certain properties which are instrumental for the convergence analysis. Assuming the initial data \(u_0 \in H^{1+\alpha }(\mathbb {R})\) $u_0\in H^{1+\alpha }(\mathbb{R})$ , where \(\alpha \in [1,2)\) $\alpha \in [1,2)$ , 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 \(\alpha \) $\alpha$ . Furthermore, we demonstrate that the Crank–Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.