Exponential memory and Caputo-Fabrizio

Exponential memory is a bounded-kernel operator, not a fractional derivative (Exponential (fading) memory). It uses TimeMemoryStepper.

This example solves

\[ \mathcal E_\lambda u(t)= \frac{1-e^{-\lambda t}}{\lambda}, \qquad u(0)=0, \]

whose solution is \(u(t)=t\).

from math import exp

from firedrake import *
from yonderdrake import (
    CaputoFabrizioOperator,
    ExponentialMemory,
    TimeMemoryStepper,
)

mesh = UnitIntervalMesh(1)
V = FunctionSpace(mesh, "CG", 1)
u = Function(V)
v = TestFunction(V)
t, dt = Constant(0.0), Constant(0.05)
source = Constant(0.0)
decay_rate = 1.7

F = (
    inner(ExponentialMemory(u, decay_rate), v)
    - inner(source, v)
) * dx
stepper = TimeMemoryStepper(F, t, dt, u)

for _ in range(20):
    target_time = float(t) + float(dt)
    source.assign((1.0 - exp(-decay_rate * target_time)) / decay_rate)
    stepper.advance()
    t.assign(t + dt)

The recurrence is exact for this linear history, so the only error left is algebraic. Construction warns that the memory starts at zero. Here the initial data are compatible.

For the parameterization used in the Caputo-Fabrizio literature, replace the marker with CaputoFabrizioOperator(u, alpha). It remains the same single exponential memory mode.