theta-scheme for solving Caputo fractional differential equations

We formulate a \(\theta\)-numerical scheme for solving Caputo fractional differential equations (FDEs) of order \(\alpha\in(0,1)\), with vector fields satisfying a standard Lipschitz continuity condition in the state variable and a Holder continuity condition in the time variable. The convergence rate is established and a numerical example is given to illustrate the theoretical results. The scheme obtained includes the explicit (\(\theta=0\)) and implicit (\(\theta=1\)) counterparts of Euler-like schemes for Caputo FDEs known in the literature as the Adams-Bashford and Adams-Moulton schemes, respectively, and essentially linearly interpolates them. For more information see https://ejde.math.txstate.edu/Volumes/2025/05/abstr.html