Susceptibility of Time-Variant Metamaterial Systems: Classical Convolution vs. ODEs with Non-local Forcing

We investigate the relationship between convolution integral representation and differential equations for disordered and time-variant systems with Gaussian-damped sinusoidal impulse response. While the impulse response satisfies a specific ordinary differential equation (ODE), we demonstrate that the general response does not satisfy this same equation. We derive an alternative ODE formulation that preserves the standard convolution form while adaptively accounting for time-variance through time and input-dependent terms. Numerical validation confirms agreement between the proposed adaptive ODE solutions and exact convolution results for various input signals. This approach enables efficient time-domain numerical schemes with controlled accuracy bounds for modeling susceptibility in metasystems with spatial disorder and external temporal control.