Help with Integrals (one of which involves erfc).

  • Thread starter 6Stang7
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  • #1
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I'm using Green's Functions for heat conduction problems, and I'm trying to solve the following integral:

Homework Statement



http://img28.imageshack.us/img28/4923/026307b169b04faa8364086.png [Broken]

Where:

http://img820.imageshack.us/img820/3742/6332938c445f4b9e8da8ba5.png [Broken]

Homework Equations


N/A


The Attempt at a Solution


I obliviously broke this up into two integrals which I'll call Integral #1 and Integral #2.

For Integral #1 I used some algebraic simplification and a u-substitution (u=4α(t-τ)) to get the following:

http://img600.imageshack.us/img600/2892/121092a52813459b8280e05.png [Broken]

Using Mathmatica, I got:

http://img856.imageshack.us/img856/719/92de0714ca1a4ec8abd7c29.png [Broken]

For which I get the following after putting in the limits of the integral and evaluate some limits:

http://img195.imageshack.us/img195/6397/26da0028e4824870aa77365.png [Broken]

For Integral #2 I again did some algebraic simplification and a u-substitution (u=α(t-τ)) to get:

http://img838.imageshack.us/img838/7506/fb9afa29d9e44617861b1a6.png [Broken]

Using Mathmatica again:

http://img194.imageshack.us/img194/2729/25901d5d37a644cbab647ec.png [Broken]

Where a=(x/2) and b=(h/k).

After doing some algebra cleanup, substituting back in the values of a and b, and applying the limits of integration:

http://img571.imageshack.us/img571/7754/c206229ff433444c9d7b31d.png [Broken]

After combining the solutions for Integrals #1 and #2 with the constants that were pulled at the start:

http://img191.imageshack.us/img191/9814/b5fd0b3ed9294586bb55efc.png [Broken]

Where fi(τ)=hT

Now, when I did dimensional analysis for x=t=0, I got units of temperature; however, when I tried to plot my solution in Matlab, I end up getting garbage. I've doubled checked my code in Matlab to make sure the solution is typed in correctly (and I am in the process of triple checking) and I have double checked my integration. So far, I can't find a mistake.

I'd prefer to do the integrals without Mathmatica, but I haven't been able to find an appropriate table of integrals. I THINK I can do Integral #1 entirely by hand using the definition of the error function, but that still leaves Integral #2, which I haven't been able to crack.

Any thoughts/help/advice?
 
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Answers and Replies

  • #2
Ray Vickson
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How can you do the integrals without knowing what is the function f_i(tau)?
 
  • #3
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fi(τ)=hT

I did state that, but it was after the final solution I got (my mistake for stating it so late).
 
  • #4
SteamKing
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When you say you are getting garbage when you try to plot, what does that mean exactly?
Have you tried to see if Mathematica evaluates erf(x) properly?
 
  • #5
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By garbage I mean that the plot doesn't make any physical sense (both in temperature values and distribution).

Mathmatica does appear to evaluate the erf(x) correctly.
 
  • #6
Matlab project

I have a homework project in matlab, Can you help me please ??
 

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