Generalized fractional differential equations for past dynamic

Well-posedness of the terminal value problem for nonlinear systems of generalized fractional differential equations is studied. The generalized fractional operator is formulated with a classical operator and a related weighted space. The terminal value problem is transformed into weakly singular Fredholm and Volterra integral equations with delay. A lower bound for the well-posedness of the corresponding problem is introduced. A collocation method covering all problems with generalized derivatives is introduced and analyzed. Illustrative examples for validation and application of the proposed methods are supported. The effects of various fractional derivatives on the solution, wellposedness, and fitting error are studied. An application for estimating the population of diabetes cases in the past is introduced.