On Numerical Methods for g-Tempered Fractional Differential Equations with Applications

This paper presents a comprehensive theoretical and computational analysis of the g-tempered L1 discretization scheme for fractional differential operators acting on Hölder spaces. While standard Volterra-type integration methods frequently suffer from severe order degradation due to initial boundary singularities and temporal delay discontinuities, the proposed L1 architecture directly linearizes the state variable over a g-warped time scale, effectively neutralizing these geometric challenges. We rigorously establish the well-definedness of the g-tempered operators, derive a formal, uniform bound on the truncation error O(τ1+β−α), which is especially relevant for Hölder-continuous functions of order β. This exact bound mathematically accommodates the limited regularity of Hölder-continuous state trajectories inherent to fractional-order systems, allowing us to introduce an asymptotic global error model to quantify the empirical order of convergence (EOC). The mathematical framework is validated through the method of manufactured solutions (MMS) across varying regularity conditions, including pre-asymptotic behavior under non-linear quadratic time warps. Finally, the discretization scheme is applied to a non-linear, delayed g-tempered SIQR epidemic model. Numerical evaluations utilizing double-mesh extrapolation confirm that the structural decoupling of the historical memory path ensures robust algorithmic stability and successfully preserves the theoretical convergence rates, even in the presence of strongly coupled bilinear interactions.