SPECTRAL DECOMPOSITION STRATEGIES FOR ANOMALOUS DIFFUSION AND IMPULSIVE BOUNDARY VALUE SYSTEMS - Nada Abdul-Hassan Atiyah (Journal for Technology and Science, 2026)
Objective: Physical models with memory or impulsive effects of non-local nature present complications mathematically. Method: To circumvent such computational challenges, a frequency domain approach is explored herein. The new algorithm allows one to deal easily with fractional derivatives and strong singularities by reducing the problem of solving the integro-differential equation to the solution of an algebraic problem. Results: Numerical analysis suggests that the transition to the frequency domain is indeed valid but exhibits a noticeable reduction in rate of convergence, which goes from O(N⁻²) in normal situations to O(N⁻ᵅ) for the anomalous case. We further consider sustained oscillations induced by impulse-like forcing. Novelty: The traditional approach using finite difference method is known to have stability issues when dealing with anomalous diffusion as well as impulses like Dirac delta functions.