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 Post subject: Transient term
PostPosted: Mon, 17 Apr 2006 02:54:02 UTC 
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Could someone give a definition of a "transient term" in the context of ordinary differential equations? I think it is defined as a term in the solution to a differential equation that approaches zero as the independent variable approaches infinity; but I am not totally sure because my textbook wasn't very clear on this.

For example, in the equation:

\displaystyle{y=\frac{\ln(x)}{x}+\frac{c}{x}}

I think that both \frac{\ln(x)}{x} and \frac{c}{x} are transient terms, since they both decay to zero as x\rightarrow\infty.

And the equation:

\displaystyle{y=-x\cos(x)+cx}
has no transient terms.

Is that correct?


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PostPosted: Mon, 17 Apr 2006 06:02:51 UTC 
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First, the terminology is generally used when the independent variable is t. The

transient term is gone "after awhile". The classical example is the damped driven harmonic oscillator.

\ddot{x} + 3\dot{x} + 2x = F\cos{wt} has as a solution

x = A\cos{wt} + B \sin{wt} + Ce^{-t} + De^{-2t}

C and D depend on the initial conditions, A and B do not (I'm too lazy to calculate them).

The exponential terms are transient, the trigonometric terms are not.


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PostPosted: Mon, 17 Apr 2006 06:12:14 UTC 
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So regarding the two examples I gave, are my guesses correct? I still can't find a precise definition anywhere...


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PostPosted: Mon, 17 Apr 2006 06:18:53 UTC 
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jinydu wrote:
So regarding the two examples I gave, are my guesses correct?



Yes, just don't use x as the independent variable.

Quote:

I still can't find a precise definition anywhere...


The "transient" isn't the problem. Give me a precise definition of "term".


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PostPosted: Mon, 17 Apr 2006 06:32:30 UTC 
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I think a term was defined as one of the elements in a sum, the same way that a factor is defined as one of the elements in a product.

So "transient term" isn't part of standard mathematical terminology?


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PostPosted: Mon, 17 Apr 2006 08:30:14 UTC 
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jinydu wrote:
I think a term was defined as one of the elements in a sum, the same way that a factor is defined as one of the elements in a product.

So "transient term" isn't part of standard mathematical terminology?


I would say it's standard, but why search for ''transient term'' and not just ''transient''. It usually is a term when it appears in mathematics, because in the driven harmonic oscillator case, it is the homoegenous (unforced) solution and as we know the solution is biult up as a sum of a homogeneous and an inhomogeneous solution. So there are two parts of the solution, one transient which we see nothing of after a while, and one steady state (particular solution in harmonic oscillator case) which is the only thing we see after a while. Generally it is ''what dies with time'' and I see no need for it to be more precise than that.


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PostPosted: Mon, 17 Apr 2006 14:34:32 UTC 
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jinydu wrote:
I think a term was defined as one of the elements in a sum, the same way that a factor is defined as one of the elements in a product.

So "transient term" isn't part of standard mathematical terminology?


This relates to ODEs.

The problem is that the solution to the ODE can be written as f(t) + g(t) where

g(t) is transient in multiple ways.


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