In this research, we explore a high-precision numerical method for non-smooth differential equations (NSDEs). Generally speaking, NSDEs can be comprehensively categorized into three main types, namely, the continuous piecewise smooth variety, the discontinuous piecewise smooth type, and the infinite derivative type. When addressing non-smooth issues of the infinite derivative type, conventional approaches typically resort to adaptive quadrature strategies, which consequently entail a substantial computational burden. With the intention of resolving this issue, we present a fractional interpolation method (FIM) that is founded on the fractional Taylor series. This particular method principally takes advantage of the distinctive characteristic that fractional interpolation is associated with infinite derivatives. Unlike conventional methods, the FIM can directly interpolate infinite derivative points without relying on an adaptive quadrature strategy, thus bringing a significant improvement in computational efficiency. Through rigorous non-smooth numerical experiments, we have demonstrated its superior performance compared to conventional high-order numerical methods and MATLAB’s built-in functions. We further confirm the practical applicability of the proposed method by applying it to piecewise smooth systems, such as dry friction systems and binary airfoil systems with free-play. This application effectively demonstrates its potency in analyzing the dynamic responses solutions of non-smooth systems.