Diffusive representations of time memory

A Caputo derivative depends on the whole history of \(u\). Diffusive methods replace that history by a fixed number of first-order memory modes,

\[ D_C^\alpha u(t)\approx\sum_{j=1}^m w_j\phi_j(t), \qquad \dot\phi_j+\lambda_j\phi_j=\dot u, \qquad \phi_j(0)=0, \]

with positive rates \(\lambda_j\) and weights \(w_j\) obtained by quadrature of

\[ D_C^\alpha u(t) =\frac{\sin(\pi\alpha)}{\pi} \int_0^\infty \lambda^{\alpha-1}\phi(t,\lambda)\,d\lambda , \qquad \dot\phi(t,\lambda)+\lambda\phi(t,\lambda)=\dot u(t). \]

Storage becomes \(O(m)\) per fractional term instead of growing with the number of time steps. The methods below differ only in how that improper integral is discretized.

Representation	Quadrature	Status	Source
BirkSong	Gauss-Jacobi after a squared Cayley map	Default	Birk and Song (2010)
Diethelm	Gauss-Jacobi after a Cayley map	Supported alternative	Diethelm (2008)
Diethelm2022	Truncated log-rate integral, trapezoidal/Simpson/Gauss-Legendre	Comparison only	Diethelm (2022), (2023)
YuanAgrawal	Original Gauss-Laguerre rule	Comparison only	Yuan and Agrawal (2002)

Prefer BirkSong or Diethelm. See use the defaults.

Refining a representation

The mode count and rate_scale are quadrature parameters of the memory integral. They are independent of the time grid. Halving dt does not improve a badly resolved spectrum, and adding modes does not fix a coarse timestep. Refine one at a time, as in Separating error sources.

Equal mode counts across representations do not imply equal accuracy. Comparisons are only meaningful against a shared analytic or high-accuracy reference. python -m benchmarks.diffusive_representations produces exactly that comparison and preserves its numerical results.