Application Domains

dfsc is strongest when a fractional model has a known linear propagator or a discrete operator whose Mittag-Leffler action can be evaluated directly. The application layer translates four recurring model classes into the same problem--algorithm--solution interface.

Domain	dfsc path	Specific advantage	Current boundary
Anomalous diffusion	Regular spectral MLSL	Batched arbitrary-time queries and trainable temporal/spatial orders without storing history	1D/2D tensor-product domains and constant orders
Linear viscoelastic or diffusion relaxation	Generalized stiffness/mass MLSL	Mass-aware projection from an existing finite-element discretization	Symmetric linear systems; matrices are supplied by the user
Network memory diffusion	Graph MLSL	A known graph propagator can be inserted directly into a neural computation graph	Undirected dense adjacency in the direct adapter
Fractional advection--diffusion	General-operator Arnoldi	Non-self-adjoint matrix-function actions without eigenvector decomposition	Periodic centered discretization and reduced argument radius at most four

Unified Application Object

Each constructor returns an ApplicationCase. It contains the configured problem, recommended algorithm, differentiable parameters, assumptions, and limitations. Calling case.solve() uses the recommended method; passing an algorithm explicitly remains possible.

case = dfsc.anomalous_diffusion_case(
    initial=lambda x: torch.sin(torch.pi * x),
    times=torch.linspace(0.0, 0.1, 8),
    alpha=torch.tensor(0.8, requires_grad=True),
    beta=torch.tensor(1.7, requires_grad=True),
    diffusivity=0.1,
)
solution = case.solve()
print(case.summary())

The catalogue is machine readable:

for profile in dfsc.application_catalog():
    print(profile["name"], profile["fit"], profile["limitations"])

Interpretation of Coverage

The four templates expand application usability around the MLSL core; they do not claim that dfsc is a complete domain simulator. Mesh generation, constitutive model selection, experimental-data calibration, nonlinear fluxes, and arbitrary memory kernels remain outside these templates. They can be coupled through supplied operators, forcing, semilinear, or neural-residual interfaces when the corresponding mathematical assumptions are justified.