@article{Merad2026, 
author = {Ahcene Merad and Abdellah Menasri},
title = {Nonlinear higher-order time-fractional equations with purely integral boundary conditions: analytical results and numerical simulations},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {15990-16007},
keywords = {time-fractional partial differential equation, purely integral boundary conditions, Caputo derivative, weak solution, existence and uniqueness, continuous dependence, nonlocal constraints, L1 method},
url = {https://www.sciopen.com/article/10.3934/math.2026658},
doi = {10.3934/math.2026658},
abstract = {This paper studies a nonlinear higher-order time-fractional partial differential equation with purely integral boundary conditions in the strip        Q    T    =  (  0  ,  1  )  ×  (  0  ,  T  ). The model involves a Caputo derivative of order    0  <  α  <  1, a variable-coefficient principal part, and a nonlinear term depending on the solution and on its first spatial derivative. The analysis is formulated for any positive integer    m under an explicit boundedness and coercivity hypothesis for the weak realization of the spatial operator on a moment-constrained space; this point is stated as a structural assumption, not as a consequence of positivity of the coefficient alone. We clarify the exact order of the variable-coefficient operator, construct a bounded lifting for the two imposed moments, give the weak duality formulation, and derive an a priori estimate with constants that do not depend on the unknown solution. Existence is obtained from a linear fractional solvability result and a fixed-point argument, while uniqueness and continuous dependence follow from a fractional energy inequality and a Mittag-Leffler version of the fractional Gronwall lemma. The numerical section is deliberately presented as a reproducible    m  =  1 validation: it includes the classical L1 finite-difference benchmark, a zero-moment forcing test, and an additional variable-coefficient nonlinear manufactured example solved by Picard iteration. The remaining extension to fully nonlinear higher-order discretizations for    m  >  1 is identified explicitly as future work.}
}