History-free spectral modelling of fractional seismic1 attenuation on fixed self-adjoint operators

Ning Hu, Shuqun Li, Chengtao Huang, Chuyang Hu

Fractional wave models represent power-law attenuation and dispersion, but direct Caputo5 time stepping retains an increasingly long memory. We isolate the fixed-operator case in6 which a prescribed positive self-adjoint semidiscrete spatial operator is queried repeatedly at7 selected output times, receiver locations, or dense-basis source weights. Its fractional tem-8 poral evolution is evaluated directly by modal Mittag–Leffler functions, using dense eigende-9 composition for moderate systems and source-dependent Lanczos matrix actions for sparse10 irregular grids. The main evidence is deliberately compact: independent same-equation11 L1/Grunwald–Letnikov receiver checks on matched two- and small three-dimensional grids12 stay below a 1.5% waveform error gate without storing a time-history array; an irregular13 14,598-cell single-source Krylov test differs by 1.445% from an extended 240-vector reference14 at m = 120 and by 0.237% at m = 160; and the reuse threshold q∗ makes the setup-versus-15 query trade-off explicit. Dense eigenbases can be reused across source weights, whereas the16 tested Lanczos route is tied to its starting source vector and reused only across times and17 receivers. A prescribed smooth variable-space-order matrix is included as a bounded diagnos-18 tic, with a discontinuous-order counterexample preventing any general continuous-operator19 1 or sharp-interface claim. The result is therefore a reproducible requested-time evaluator20 and enabling computational gateway for richer fixed-operator fractional attenuation experi-21 ments, not a universal fractional-wave solver, full-waveform-inversion acceleration, field-scale22 production code, or non-self-adjoint absorbing-boundary method.Reproducibility materials are archived on Zenodo at https://doi.org/10.5281/zenodo.21486072.