A MODIFIED NEWTON–KRYLOV APPROACH FOR SOLVING NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

Nonlinear fractional differential equations (FDEs) arise routinely in viscoelasticity, anomalous diffusion, and control models, yet their discrete counterparts produce Jacobians that are dense, ill-conditioned, and expensive to invert as the fractional order departs from unity. This paper reports an empirical evaluation of a modified Newton–Krylov (mNK) scheme in which the linear correction step is solved using a preconditioned GMRES iteration built on a shifted Grünwald Letnikov operator, rather than a full Newton step recomputed at every outer iteration. Five benchmarks nonlinear FDEs, spanning fractional orders α between 0.3 and 1.8 and drawn from Caputo and Riemann–Liouville formulations, were discretised using an L1/L2 scheme and solved on mesh sizes ranging from 64 to 2048 nodes. Outer-iteration counts, residual decay, wall-clock time, and memory footprint were recorded and benchmarked against classical Newton, Newton GMRES without the fractional preconditioner, and an Adomian decomposition baseline. The modified scheme reduced average outer iterations by 34–47% and CPU time by roughly 40% relative to unpreconditioned Newton–Krylov, with the advantage widening as α approached the diffusion-dominated regime near 1.8, where classical Newton frequently stagnated. Error norms remained within 10⁻⁶ to 10⁻⁸ across all test cases. These findings support the abstract's central claim that coupling a fractional-order-aware preconditioner with inexact Newton iteration yields a numerically robust and computationally economical route to nonlinear FDE solutions, a conclusion the paper returns to and substantiates further in its closing discussion.