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From: Osher Doctorow on 14 May 2010 01:51
From Osher Doctorow

The exponential growth/decay subcases of the Riccati Differential
Equation have respective solutions y = y(0)exp(kt), y = y(0)exp(-kt),
k > 0.

If tau is an earlier time than t, then a Memory version of the above
would be:

1) y = y(0)exp(k(t - tau)), y = y(0)exp(-k(t - tau)), k > 0

Notice that in (1), the first equation acts to slow down the
exponential growth, and it was pointed out in previous posts in this
thread that Memory tends to be slower than exponential growth.

To develop an equation with slower than exponential growth, a more
general procedure appears to be using two times, t and u. Then the
expression:

2) y = exp(t) - exp(u) has slower growth than y = exp(t). Similarly
for:
3) y = exp(kt) - exp(ku), k > 0, compared with y = exp(kt).

Notice that:

4) Dt(y) = kexp(kt) for y given by (3).
5) Du(y) = -kexp(ku) for y given by (3).

Then:

6) Dt(y) + Du(y) = ky, for y defined by (3) (left expression of (3)).

Let us define:

7) Dt+u(y) = Dt(y) + Du(y) (where the left hand side for clarity can
be written D[t+u](y)).

Then:

8) Dt+u(y) = ky generalizes the Riccati Differential Equation subtype
dy/dt = ky, k > 0, involves 2 times t and u ("independent" times), and
involves Memory with solution y = exp(kt) - exp(ku).

We also have from (8):

9) y is nonnegative in (8) if 0 < u < t (so u acts as a "prior to t"
but otherwise independent time variable in the nonnegative case).

These equations are simpler than either Delay-Differential or Volterra
Integral Equations, both in form and in solutions and derivation of
solutions, while revealing better in some way the deeper structure of
Memory and Riccati Differential Equations (generalized).

Osher Doctorow
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