TY  - JOUR
AU  - Batiha, Iqbal M.
AU  - Albteiha, Sharaf Aldeen O.
AU  - Al-Khawaldeh, Hamzah O.
AU  - Momani, Shaher
PY  - 2026
TI  - Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems
JO  - AIMS Mathematics
SP  - 483
EP  - 510
VL  - 11
IS  - 1
AB  - 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    α  ∈    [  0.5  ,  1  ] and    q  ∈  (  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.
UR  - https://doi.org/10.3934/math.2026021
DO  - 10.3934/math.2026021