# How to extend the code to a system with delay?

Posted 10 months ago
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 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:
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Posted 10 months ago
 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.
Posted 10 months ago
 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.
Posted 10 months ago
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