An error-certified matrix Mittag–Leffler perturbation method for weakly nonlinear fractional systems

Salas A.H.; Altamirano G.C.; Martínez H L.J. · Frontiers in Applied Mathematics and Statistics (2026) · DOI: 10.3389/fams.2026.1899674

Resumen

Introduction – Weakly nonlinear fractional systems combine nonlocal memory with nonlinear response. A reliable approximation must therefore control both perturbation truncation and numerical evaluation of the memory integral. Methods – We develop an error-certified matrix Mittag-Leffler perturbation method for Caputo systems of the form D C t α X ( t ) = A X ( t ) + ε G ( t, X ( t ) ),    X ( 0 ) = X 0,    0 < α < 1. The exact matrix propagator (Formula presented) is retained, while nonlinear corrections are transported by the kernel (Formula presented). We prove an explicit first-order truncation estimate, a general analytic-series remainder bound, and an a posteriori residual-to-solution estimate. A power transformation removes the weak endpoint singularity, permitting standard Gauss-Legendre quadrature. Validation uses an exact manufactured problem, a fractional Duffing-type state-space system, and a nonlinear fractional logistic equation, with refined ABM-PECE and implicit Grünwald-Letnikov solutions as independent benchmarks. Results – The manufactured problem gives a maximum implementation error of 6.025 × 10−7. For the Duffing case with α = 0.85, ε = 0.10, and T = 3, the actual L ∞error is 2.131 × 10−3and the Volterra residual is 2.176 × 10−3. The observed perturbation orders range from 1.991 to 1.918, confirming the predicted O(ε2) behavior. Fractional-order, long-time, step-refinement, and CPU studies delimit the practical range of the method. Discussion – The method is particularly attractive for parameter studies in which the same linear fractional propagator is reused. Its limitations include strong nonlinearities, long memory horizons, conservative global bounds, and the matrix-function cost for large systems.

Instituciones: Universidad Nacional Autónoma de Chota