Help with an elimination and integration...

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masher
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Help with an elimination and integration...

Postby masher » Thu Mar 15, 2012 6:02 am UTC

I'm having a bit of trouble following some maths in a paper I'm reading:

There are two equations defined as:

[math]\frac{d\eta}{dt} = - \psi^n[/math]
[math]\frac{d\eta}{dt} = \frac{1-\psi}{\sigma^2} \frac{1}{\frac{dp(\eta)}{d\eta}}[/math]

The paper then says:
As usual, we may now proceed to eliminate [imath]\psi[/imath] by combining the two equations and thus obtain expressions for the rate which can then integrated to yield a relationship between [imath]\eta[/imath] and [imath]t[/imath].


The paper gives an example for n=1 being:
[math]t=(1-\eta) + \sigma^2 p(\eta)[/math]

For the life of me, I can't figure out how to get past the first stage of combining the equations to get rid of [imath]\psi[/imath]...

Any ideas/help?

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jestingrabbit
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Re: Help with an elimination and integration...

Postby jestingrabbit » Thu Mar 15, 2012 7:30 am UTC

Using just the first equation, can you write down an expression for psi that you can sub into the second?

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Proginoskes
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Re: Help with an elimination and integration...

Postby Proginoskes » Thu Mar 15, 2012 8:13 am UTC

masher wrote:I'm having a bit of trouble following some maths in a paper I'm reading:

There are two equations defined as:

[math]\frac{d\eta}{dt} = - \psi^n[/math]
[math]\frac{d\eta}{dt} = \frac{1-\psi}{\sigma^2} \frac{1}{\frac{dp(\eta)}{d\eta}}[/math]

The paper then says:
As usual, we may now proceed to eliminate [imath]\psi[/imath] by combining the two equations and thus obtain expressions for the rate which can then integrated to yield a relationship between [imath]\eta[/imath] and [imath]t[/imath].


I'd set the two equations equal to each other, multiply both sides by [imath]-\psi^n \sigma^2 \,dp(\eta)[/imath] to get:
[math]\sigma^2 \,dp = \psi^{n}(\psi-1)\,d\eta.[/math]
Then
[math]\sigma^2 p = \int \sigma^2\, dp = \int \psi^n(\psi-1)\,d\eta = \psi^n(\psi-1)\eta + C[/math]
for some constant C. This assumes, of course, that n is a constant w.r.t. [imath]\eta[/imath] and [imath]\sigma[/imath] is a contant w.r.t. p. Otherwise, there are more equations to post.

The paper gives an example for n=1 being:
[math]t=(1-\eta) + \sigma^2 p(\eta)[/math]

For the life of me, I can't figure out how to get past the first stage of combining the equations to get rid of [imath]\psi[/imath]...

Any ideas/help?

masher
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Joined: Tue Oct 23, 2007 11:07 pm UTC
Location: Melbourne, Australia

Re: Help with an elimination and integration...

Postby masher » Thu Mar 15, 2012 10:55 am UTC

n and sigma are constants.

The best I can do is the following:

Equate the expressions:
[math]- \psi^n = \frac{1-\psi}{\sigma^2} \frac{d\eta}{dp}[/math]
[math]\sigma^2 dp = - \frac{1-\psi}{\psi^n} d\eta[/math]

We know that [imath]d\eta = - \psi^n dt[/imath].
[math]\sigma^2 dp = (1-\psi) dt[/math]

Integrate both sides
[math]\sigma^2 p = (1-\psi)t + C[/math]

We also know that [imath]\eta = - \psi^n t + D[/imath], and if we take n=1 (as in the given example), we find that [imath]\psi = \frac{\eta-D}{t}[/imath].

Substitute in, and get
[math]\sigma^2p = t + \eta - D + C[/math]

and finally
[math]t = -C - \eta + \sigma^2 p[/math]

which is a little different to the given example (unless C = -1)...

masher
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Location: Melbourne, Australia

Re: Help with an elimination and integration...

Postby masher » Thu Mar 15, 2012 10:15 pm UTC

I can also do this:

Given that [imath]\frac{d\eta}{dt} = - \psi^n[/imath], we can see that [imath]\eta=- \psi^n t + C[/imath], and consequently, if n=1, [imath]\psi = \frac{C-\eta}{t}[/imath].

From this, it follows that
[math]\frac{d\eta}{dt} = \frac{1-\frac{C-\eta}{t}}{\sigma^2} \frac{d\eta}{dp}[/math]

Canceling [imath]d\eta[/imath] and inverting both sides of the equation, we get
[math]dt = \frac{\sigma^2}{1-\frac{C-\eta}{t}} dp[/math]
[math]\left( 1-\frac{C-\eta}{t} \right) dt = \sigma^2 dp[/math]

Integrating both sides, we find:
[math]t - (C-\eta)\ln(t) = \sigma^2 p + D[/math]


which is different to both my previous result, and the result given in the paper...

masher
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Re: Help with an elimination and integration...

Postby masher » Fri Mar 16, 2012 1:40 am UTC

Got it!

Taking my first solution:
[math]t = -C -\eta + \sigma^2 p[/math]

I can then use some boundary conditions I found implied in one of the figures: at t = 0, [imath]\eta =1[/imath] and p = 0. It follows from this that C does, in fact, equal -1.

Finally, in the case of n = 1,
[math]t=1-\eta + \sigma^2 p[/math].


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