<p>This study focuses on the space fractional advection-dispersion equation. We develop a stable second-order fractional cubic spline minimum residual method (FCS-MRM) based on second-generation reproducing kernel functions. To the best of our knowledge, this is the first attempt to construct fractional cubic spline functions using reproducing kernel theory. This approach significantly simplifies the function construction process and enhances flexibility. By reformulating the equation under homogeneous boundary conditions, a unified theoretical framework is established. Time discretization is achieved via a second-order finite difference scheme, while the spatial solution is represented using fractional cubic spline basis functions. A residual functional is minimized to obtain the approximate solution. Convergence, error behavior, and stability are rigorously analyzed and proven. Numerical experiments confirm the accuracy and robustness of the proposed method. In a series of benchmark problems, the proposed method demonstrates superior performance compared to classical approaches. These include integer-order cubic splines, finite difference schemes, alternating direction implicit methods based on Crank-Nicolson, and reproducing kernel particle methods. The method consistently yields significantly lower errors.</p>