Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems

Motivated by the challenges faced by standard methods in solving nonlinear fractional and q-fractional models with strong memory effects, this study develops numerical approaches capable of handling these complex behaviors more effectively. The proposed techniques are tested on four representative fractional and q-fractional initial value problems for several values of the order $\alpha \in$ $[0.5,1]$ and $q\in (0,1)$. In particular, the major aim of this work is to propose two generalizations of the higher-order Taylor method: The first one is the fractional Taylor method, and the second one is the q-fractional Taylor method. These methods will then be used to find approximate solutions for several fractional and q-fractional initial value problems. Numerous numerical comparisons will be performed to verify the effectiveness of our proposed generalizations.