@article{Batiha2026, 
author = {Iqbal M. Batiha and Sharaf Aldeen O. Albteiha and Hamzah O. Al-Khawaldeh and Shaher Momani},
title = {Generalizations of higher-order Taylor method: Fractional and q-fractional approaches for initial value problems},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {1},
pages = {483-510},
keywords = {fractional calculus, fractional q-calculus, fractional Taylor method, fractional q-Taylor method, numerical solutions},
url = {https://www.sciopen.com/article/10.3934/math.2026021},
doi = {10.3934/math.2026021},
abstract = {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.}
}