Application of the Adams-Bashfort-Mowlton Method to the Numerical Study of Linear Fractional Oscillators Models

The paper formulates the FLO Cauchy problem with Caputo derivatives 1<β<2 and 0<γ<1 and analyzes two solvers: an explicit nonlocal finite-difference scheme (ENFDS) and a fractional Adams–Bashforth–Moulton (ABM) predictor–corrector. It states and cites proofs that ENFDS is first-order convergent under a stepsize restriction τ≤τ0 and is conditionally stable (Theorems 1–2), and that ABM attains a global error O(τ^{1+min(σ1,σ2)}) after splitting with σ1=γ and σ2=β−γ (Theorem 3). These claims and the concrete step condition agree with the candidate’s derivations and assumptions, which add a standard product-integration viewpoint and a discrete fractional Grönwall step for stability. The paper’s numerical Example 1 (with exact solution x=t^3) shows ABM’s observed order ≈1.9 and ENFDS ≈1, matching the candidate’s reported trends. The paper’s Example 2 shows a case where ABM’s advantage diminishes, with both methods’ “computational accuracy” approaching 1; the candidate explains this behavior can depend on start-up accuracy and regularity, and notes that with a second-order start (not used in the paper) ENFDS can recover second-order in the β=2, λ=0 limit, which is compatible with the theory but beyond the paper’s scope. Overall, both presentations are consistent on the main theoretical points and empirical trends, but proceed via different proof sketches and implementation details. Key places in the paper supporting this assessment include the problem statement and ENFDS construction, the step restriction and stability/convergence theorems, the ABM construction and error theorem, and the numerical tables for Examples 1–2. See the paper’s statement of the problem and ENFDS discretization, the convergence/stability theorems, ABM formulation and global error result, and Example 1/2 results, respectively: .