A Two-Sweep Fractional Adams Predictor–Corrector Method for Nonlinear Caputo Initial-Value Problems

Abstract

We give a self-contained analysis of a two-sweep, defect-corrected fractional Adams predictor–corrector (P(EC) E, “PECECE”) method for nonlinear Caputo initial-value problems of order . The scheme is the classical product-integration Adams–Bashforth predictor followed by two Adams–Moulton fixed-point corrections that share a single, precomputed history convolution; the second correction therefore costs exactly one additional evaluation of the nonlinear right-hand side and no new  history work. This construction is a special case ( ) of the general P(EC) E family Diethelm, Ford and Freed described in their foundational papers, and we make that lineage explicit rather than let the reader assume otherwise. What this paper contributes is not the second corrector sweep itself but three things that, as far as we can tell, no one has put together for this specific case before: (i) an explicit nonlinear contraction bound  that serves as a practical, computable diagnostic for whether a second sweep is worthwhile; (ii) a Mittag–Leffler nonlinear stability estimate with the contraction factor tracked through the constant; and (iii) a reproducible, cost-aware numerical protocol, including, for the first time in this line of work, an explicit measurement of the experimental order of convergence (EOC) against the theoretical bound . That measurement turns up a discrepancy worth reporting in its own right: for a smooth manufactured solution, the observed EOC is close to 2 across all tested , exceeding the theorem’s guarantee most sharply at low , consistent with grid-point superconvergence for regular data rather than any flaw in the bound. We report this honestly, quantify the extra computational cost of the second sweep, and confirm the theorem’s sharpness directly with a genuinely weakly singular companion test (Section 7.4), which shows the observed order degrading to the expected 2α once the regularity clause is actually violated. What comes out of this is a narrow but rigorous, fully reproducible baseline — not a claim to new algorithmic territory.