A Boundary-Adapted Legendre–Galerkin Method for Nonlinear Caputo Reaction–Diffusion Equations with Non-Local Integral Boundary Conditions

This paper studies nonlinear time-fractional reaction–diffusion equations with Caputo memory and non-local integral boundary conditions on a bounded interval. The aim is to formulate a boundary-compatible well-posedness framework and to construct a high-order temporal approximation that can be coupled with a constraint-preserving spectral spatial discretization. The analytical part proves boundedness of the non-local boundary functionals, states compatibility assumptions, and introduces a finite-dimensional nondegeneracy condition for an explicit polynomial lifting. Under a sectorial non-local elliptic realization and a global Lipschitz reaction term, existence, uniqueness, stability, and continuous dependence of mild solutions are obtained by fractional resolvent estimates and fractional Gronwall inequalities. The main novelty is the combined construction of an explicit polynomial lifting for integral boundary constraints, a constraint-preserving Legendre–Galerkin basis, and a high-order Beta-window temporal quadrature together with a discrete stability condition that accounts for sign-changing weights. The numerical evidence shows high-order behavior for smooth Caputo benchmarks, accurate enforcement of the non-local boundary constraints, and improved accuracy over the classical L1 approximation in the reported tests. The stability discussion identifies the discrete coercivity condition required for the sign-changing Beta-window weights.