Nonlinear higher-order time-fractional equations with purely integral boundary conditions: analytical results and numerical simulations

This paper studies a nonlinear higher-order time-fractional partial differential equation with purely integral boundary conditions in the strip $Q_T=(0,1)\times (0,T)$. The model involves a Caputo derivative of order $0<\alpha <1$, 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.