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.

We study an explicit finite difference approximation for the one-dimensional heat equation with purely integral conditions. The integral constraints are discretized by the trapezoidal rule, which yields explicit formulas for the boundary values at each time level and leads to a dense iteration matrix for the interior unknowns. The scheme is written in matrix form, its local truncation error is estimated, and a stability-convergence statement is established under a natural power-boundedness assumption on the full iteration matrix. Numerical experiments are reported for both stable and unstable time steps. In particular, a fixed-final-time convergence study confirms the expected first-order behavior with respect to the time step when $k$ and $h^2$ are refined simultaneously. The paper also documents the larger errors observed near the first and last interior nodes, a characteristic feature of the boundary reconstruction induced by the integral conditions.