Work

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

Year: 2026

Type: article

Abstract: This paper develops an analytical and computational framework for nonlinear time-fractional reaction--diffusion equations with non-local integral boundary conditions. The time derivative is taken in the Caputo sense of order 0< α < 1, so that the model incorporates hereditary memory while preserving classical initial data. The boundary conditions couple endpoint values with weighted spatial averages of the solution, producing a problem with both temporal and spatial non-locality. The first contribution is a boundary-compatible formulation of the problem. We prove that the non-local boundary functionals are bounded on the natural Sobolev space and introduce a finite-dimensional nondegeneracy condition that guarantees the existence of an explicit polynomial lifting. This lifting reduces the original nonhomogeneous non-local boundary constraints to a homogeneous boundary-adapted problem. Under sectoriality and global Lipschitz assumptions on the reaction term, we establish existence, uniqueness, stability, and continuous dependence of mild solutions by means of fractional resolvent estimates and fractional Gronwall inequalities. The second contribution is a high-order Beta-window, equivalently Chebyshev–Lobatto cubic, approximation of the Caputo derivative. The Caputo memory integral is decomposed into local history windows; on each interval the solution is represented by a cubic interpolant at Lobatto points, and the weakly singular kernel is integrated through explicit incomplete-Beta weights. This construction preserves the memory structure of the Caputo operator and gives a high-order temporal approximation for smooth solutions. The numerical part combines the proposed temporal approximation with a boundary-adapted Legendre--Galerkin representation. Five experiments validate the method: scalar Caputo tests, a linear manufactured fractional reaction–diffusion problem, a nonlinear manufactured balance, a non-local boundary lifting test, and a direct comparison with the classical L1 scheme. For ρ =0.65, the proposed approximation exhibits observed rates near 3.3, while the L1 method approaches the expected rate 2 − ρ =1.35. The non-local boundary residuals remain at machine precision. These results show that the proposed framework is accurate, stable, and well adapted to fractional reaction–diffusion equations with integral boundary constraints.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract10070434

Authors: Weaam Alhejail, Kawther Alsaadi, Álvaro H. Salas S.

Institutions: Princess Nourah bint Abdulrahman University, Universidad Nacional de Colombia

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funder: Princess Nourah Bint Abdulrahman University