Re: Solving a Non-Linear Delay Differential Equation

"Patoche" <patrickt@xxxxxxx> writes:

Hi there,

Is there an analytical solution to the following Non-Linear Delay
Differential Equation (DDE) ?

(1) b*y(t) + (1/a) * ( y'(t) )^a - y(t-1) = 0

I am looking for a function y(t) defined on the positive real line and
that solves equation (1).

The case a = 1 is standard in the theory of linear DDEs. I am assuming
a is strictly between 0 and 1.

If b = 1, there are solutions y(t) = c and y(t) = c + d t where
d = (-1/a)^(1/(1-a)).
Of course the latter are not real for most values of a in that interval.

Assuming y(t) is twice differentiable, then (1) implies

(2) b*y'(t) + ( y'(t) )^a * y''(t) / y'(t) - y'(t-1) = 0

Plugging (2) into (1) yields (3) :

(3) b*y'(t) + a * y''(t)/y'(t) * ( y(t-1) - by(t) ) -
y'(t-1) = 0

Provided y(t) and y'(t) are strictly monotonic, then a solution of (1)
should also solve (3). I was rather hoping that a solution of (3)
would help in solving (1).

A solution of (3) may be found, of the following form y(t) = constant
* exp(s*t), where s is equal to s = - ln(b). Equivalently, y(t) =
constant * (1/b)^t. Plugging y(t) = exp(s*t) into (3) yields: b*s +
a*s*( exp(-s)-b ) -s*exp(-s) = 0, or equivalently (1-a)*(b-exp(-s)) =
0, so provided a is strictly greater than 1, then s must satisfy b =
exp(-s), or s = -ln(b), as stated above.

However, the function exp(s*t) solves (3) but does not solve (1).
Mistake? Should I not expect a solution of (3) to solve (1) ?

It's quite obvious that c exp(s t) can't solve (1) if a <> 1.
So (3) is necessary but not sufficient for a solution.

Does anyone know how to solve equation (1) ?

Any leads will be appreciated. Many thanks!

Patrick.

--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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