The present work is concerned with the well-posedness and efficient
numerical algorithm for a terminal value problem with a generalized
Caputo fractional derivative. We investigate the existence and
uniqueness of the solution of the terminal value problem, and consider
the continuous dependence of the solutions on the given data. To
illustrate our theoretical results, we present a one step algorithm for
solving the considered problems. Some numerical examples are shown to
illustrate the theoretical results and the efficiency of the numerical
method.