Message Boards Message Boards

GROUPS:

How to extend the code to a system with delay?

Posted 10 months ago
1443 Views
|
4 Replies
|
0 Total Likes
|

I tried to find the solution of a fractional order dynamical system ( here x(t) and y(t) are functions of time alone) using an iterative procedure and it was working good, but I am in a situation to apply the same algorithm to the case of a fractional order dynamical system with delay (here x and y are of the form x(t - 0.3) and y(t - 0.3)) . So in such case how to modify the above code.

Interested people please have a look on the attached notebook for reference. I feel thankful for any suggestions.

Attachments:
4 Replies

I need to solve the following Fractional Order Delay Differential System. If possible could anyone please edit the attached notebook for the following Fractional Order Delay Differential System:

D^\[Alpha] x (t) = z + (y (t - \[Tau]) - a)*x;

D^\[Alpha] y (t) = 1 - b*y - (x (t - \[Tau]))^2;

D^\[Alpha] z (t) = -x (t - \[Tau]) - c*z;

where,

a = 3; b =0.1; c = 1, [Tau] = 0.35 , and x(0) = 0.1; y(0) =4; z(0) = 0.5 and [Alpha] = 0.90.

In my attached notebook Mathematica code was written for a Fractional Order Differential System and it works good but i am blank on how to extend it to a delay system.

Please help me.

Dear Vijayalakshmi Palanisamy,

editing a notebook is definitely not the responsibility of this site! But maybe I can make some suggestions: When you have a differential equation with a time delay you can rewrite this into a "regular" differential equation by replacing the delayed terms by its series expansion:

$ x(t-\tau)=x(-\tau )+t x'(-\tau )+\frac{1}{2} t^2 x''(-\tau )+\frac{1}{6} t^3 x^{(3)}(-\tau ) + \cdots $

This is as trivial as fundamental, and it shows the true nature of the problem: Time delay makes it a differential equation of (theoretically) infinite order.

What might help is to make an integral transform (e.g. FourierTransform) of the equations: Concerning delay you get:

$ f(t-\tau) \longrightarrow F(\kappa)\mbox{e}^{-\imath 2\pi\kappa \tau} $

and any fractional order can also quite nicely be expressed because we always have (for any $n$):

$ f^{(n)}(t) \longrightarrow (+2\pi\imath\kappa)^n F(\kappa) $

OK, these are just some thoughts from a non(!)-mathematician.

Regards -- Henrik

Posted 10 months ago

Crossposted here.

Welcome to Wolfram Community!
Please make sure you know the rules: https://wolfr.am/READ-1ST
Please next time link your post to the duplicated one from MSE site.

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract