How to extend the code to a system with delay?

Questions
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1

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.

threedimension.nb (22.1 KB)

2

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.

3

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

4

Crossposted here.

5

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