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Physical model of (infinite) tetration=(NON-isotropic) turbulence
#9
I will give it a try:

Let us take first order homogenous delay differential equation without dependence of derivative of solution on at the same moment of time :



Here T is delay, which shows how values of solution T moments in past ( or future if T<0) impact derivative of solution at current moment .

Let us introduce new function such that :

then

As is known from literature, solution of such delay equation is given by :



So

Where is a k-th branch of Lambert W function.

To reach the solution I am looking for, of form:



Coefficients have to be:

so



This can be expressed as infinite product, with each term of sum as exponent of one multiplier in this infinite product.

From here, obviously:



so





Now if we look back to original delay equation, this means that solution of a form:

is one of particular solutions corresponding to particular branch of W function of a homogenous logarithmic first order differential equation with IMAGINARY DELAY .



corresponds to Imaginary delay .

We can generalize this to imaginary delay . (Not sure if + sign is OK).

Then we have particular solutions of :



In form:



Where



I hope there are not too many mistakesSad

I think I could not choose coefficients so arbitrary, it must have an impact on initial conditions or so called preshape function for times which I do not know how to calculate.

Next question is how these logarithmic differential delay equations (and thus processes they represent) with imaginary delay have to be nested to produce tetration and , perhaps, turbulence. In case of turbulence, the statistical character of processes will have to be added, somewhere, so that we deal with mean values , distributions and structure functions. (probably, In projective space as that is the space of turbulent time).
Ivars
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Messages In This Thread
RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 03/17/2009, 08:52 PM

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