Numerical Analysis of Prabhakar Fractional Differential Equations: Error Bounds, Stability, and Green's Function Methods

Prabhakar fractional PDEs require discretization of singular Mittag-Leffler kernels. Thispaper establishes three results: (1) Lemma proving singularity extraction preserves O(∆x4)convergence via analytical log-integral and Simpson quadrature. (2) Theorem with composite error bound O(∆t2 + ∆x4 + exp(−ρnmax)). (3) Stability validation across 125 parametercombinations (α × β × γ grid). Tests include grid refinement (Zeng comparison < 0.1%),discontinuous data (2nd-order), and 20 manufactured solutions across parameter space. Independent verification of Karimov et al. (2025) Green’s function via classical heat equationlimit (agreement 4.0 × 10−10) confirms construction.