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Andrew_

Guest

A difficult integral

Hello there,

I have this very difficult integral to solve. D is a constant. Taw and Zetta can be considered as parameters with x and t being the 2 independent variables. The answer must be a function of both zetta and taw , but it seems impossible to get. If it's possible , can someone please evaluate the integral using Maple or similar software ? I dont have any installed on my computer and would really appreciate it. If someone has a method to analytically solve this, that would be even better.

Thank you :-)

Andrew_

Guest

Re: A difficult integral

I'm sorry about the confusion but the last variable zetta must be replaced by x. I made a mistake in writing it down.

This is the correct integral to solve :

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Andrew_

Tau needs to be defined a little better. What values can it take? For instance

Is it, complex, real?

Last edited by bobbym (2009-08-20 09:54:11)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Andrew_

Guest

Re: A difficult integral

Hello, I'm sorry for constantly changing the integral but it seems I keep miswriting a variable. It's quite perplexing actually. Taw is time , always positive. Zetta , however, can be negative , it belongs to R. All variables are real. D is a constant which is always positive as well.

This integral comes directly from the solution to the inhomogenous parabolic diffusion equation , I seek a solution in the asymptotic limit ( w.r.t time taw ) , however, solving the integral exactly is prefered.

Thanks, bobbym

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Andrew_;

Just trying to understand but doesn't t represent time in these type of integrals.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Andrew_

Guest

Re: A difficult integral

Hello,

I'm not quite sure I follow. The variable 't' represents dimensionless time while 'x' represents dimensionless position. But why would this matter ? Did you try to solve it using a math software ?

Thanks,

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Yes, both Mathematica and Maple are demanding more info about how the constants relate.

Taw is time , always positive

I only asked because you said tau is time.

Also the inner integral contains

implying that tau must be greater than t

Last edited by bobbym (2009-08-20 10:57:23)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Andrew_

Guest

Re: A difficult integral

The variables zetta and taw can taken as constants ,  so what's there more to relate ? All variable are independent.

Is it possible that you just solve the integral with respect to x , keep t and taw constants outside the integrals  or just define a new constant in terms of them , let this constant be positive. Then solve the single integral with respect to x, the result must be a function of zeta and t / taw , which can then be integrated w.r.t time.

Here is how I got the solution from the PDE below :

Ofcourse the variables are different here and the constants a lot more , but the integral is still the same.

Andrew_

Guest

Re: A difficult integral

Yes, both Mathematica and Maple are demanding more info about how the constants relate.

Taw is time , always positive

I only asked because you said tau is time.

Also the inner integral contains

implying that tau must be greater than t

Yes definitely 't' can never be greater than taw as it obviously varies between 0 and taw.

Thanks for your help,

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Andrew;

The picture you show in post # 8 does not match the inner integral in post #4. This symbol

you have called x in post #4, this is not correct.

Last edited by bobbym (2009-08-20 11:14:15)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Andrew_

Guest

Re: A difficult integral

Yeah I'm sorry for making you suffer with me in this. I must have changed the variables before. Here it is again,

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Andrew_;

No problem, I just need some time to fail again.

Last edited by bobbym (2009-08-20 12:14:20)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Andrew_;

This is your integral in rearranged form.

It can be reduced to this under certain very limited conditions. I have replaced D with c to avoid confusion.

This can be simplified to:

Many constants can be moved to the left of the integral operator.
Looks like a big mess but actually progress has been made.
A closed form could not be gotten for the above integral. But if we integrate from 0 to 1 instead of 0 to tau it can be done.

Now calling:

Now from 0 to 2.

Now from 0 to 3.

Now from 0 to 4.

Now from 0 to 5.

Now from 0 to 6.

.
.
.

It is possible from this to deduce a general form for 0 to n (tau).

Last edited by bobbym (2009-08-20 22:33:01)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Andrew_

Guest

Re: A difficult integral

Thanks a lot bobbym for your time and effort. It will take some time now to see what you've done and then check if this works as a solution to the PDE.

Thanks again,

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Andrew_;

Remember, this particular solution is only valid for a very particular relationship between the constants. If you need it I can work out the solution to the integral based on the 6 examples I have provided.

Also I have not attempted to clean up any of the answers given above. The reason for that will become clear later.

Last edited by bobbym (2009-08-20 22:34:41)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Chris Cox

Guest

Re: A difficult integral

I am not sure if this would work, but you could attempt to apply the Leibniz rule of integration (a.k.a. differentiating under the integral sign)
It states that:

(d/dt)∫f(x,t)dx = ∫(∂/∂t)f(x,t) + f(b,t)(∂b/∂t) - f(a,t)(∂b/∂t)

*note* Each of the integrals are from a to b, hence the last two parts of the relation ship. If a and b are constants, the last two parts disappear.

I am not sure how well it would workor even if it works with iterated integrals, though.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A difficult integral

Hi Chris;

Welcome to the forum.

I am not seeing how that is going to simplify andrews problem which in all probability is not doable in terms of elementary functions.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.