Thursday, January 15, 2009

Exact Verhulst Solution

Date: Thu, 8 Jan 2009 14:54:12 -0700 (MST)
From: Willy Hereman
To: John Stockwell
Cc: Doug Baldwin
Subject: Re: Chris Liner's paper using the Verhulst eq

John and Doug,

I read Liner's article (attached for you Doug).  Interesting!  I also did some work on the Verhulst equation, i.e. eq. (1) in his paper:

q'(t) = a_1 q - (a_1/a_2) q^2,

where a_1 is the intrinsic growth rate and a_2 is the saturation level
(also called the carrying capacity).

Eq. (1) has an exact solution which can be computed by separation of variables or by treating (1) as a Bernoulli equation. The solution is then represented as a rational expression involving an exponential function. That form of the solution can be found in almost any book on ODEs.

However, by looking at Liner's curve of the derivative, q'(t), in Fig. 1 (b), it came to me that the exact solution might be expressible in terms of a tanh function for its derivative is then sech-squared (a bell shaped, not to be confused with a true Gaussian curve, although they look alike).

Several years ago, Douglas Baldwin and I desigend a Mathematica program that automatically computes the exact tanh solutions of ODE and PDEs.  So, I tried our program and here is the nice closed form solution of Eq. (1) produced by the code:

q(t) = (1/2) a_2 { 1 + tanh[ (1/2) a_1 t + delta ] }

and its derivative

q'(t) = (1/4) a_1 a_2 sech^2 [ (1/2) a_1 t + delta ]

These are the exact mathematical expressions of the curves Liner
plotted in Fig. 1 (a) and (b), respectively.

Well, I learned something today.  I had not realized up to now that Verhulst's logistic equation had a simple tanh solution!

I have attached the Mathematica notebook with the result obtained
by our PDESpecialSolutionsV2.m code (the code is also attached).



ps (5-apr-2011)

The delta in the solution is an arbitrary constant. It is equivalent to writing the solution as

q(t) = (1/2) a_2 { 1 + tanh[ (1/2) a_1 (t - t_0) ] }

for an arbitrary t_0, i.e., an initial value for time t. The remaining two constants are a_1 and a_2.



Dr. Willy A. Hereman, Professor
Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, CO 80401-1887, U.S.A.


The article discussed here is Seismos: To peak or not to peak (Liner, 2008, The Leading Edge 27, p.610), and this is the figure:

Nonlinear waves

A bit late for this post, but I wanted to get it up here anyway to acknowledge the kindness of the sender.  The original email date was 10/22/2008.

Dear Chris,

I hope this e-mail is finding you well.

I read with interest your TLE column this month on harmonics.

I find it to be a fascinating topic.

Note that not all sources of non-linearity are due to imbalances in up and down strokes.

In water for example, non-linearity can come from change of velocity with pressure.

The higher the pressure, the higher the velocity; so when you send a sine function through water (with enough energy to affect velocity) peaks travel faster than troughs.

Thus the sine function gradually transforms into a see-saw function.

The Fourier transform of a see-saw function is a series of spikes (harmonics) with an amplitude following 1/n.

(Note that water also suffers from imbalance of up and down strokes: it is easier to push water than to pull it (a hard pull creates a vacuum in a phenomenon known as cavitation).)

Best regards,

Guillaume [Cambois]