Help with Integrals (one of which involves erfc).

1. Mar 29, 2013

6Stang7

I'm using Green's Functions for heat conduction problems, and I'm trying to solve the following integral:

1. The problem statement, all variables and given/known data

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

Where:

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

2. Relevant equations
N/A

3. 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.

Last edited by a moderator: May 6, 2017
2. Mar 29, 2013

Ray Vickson

How can you do the integrals without knowing what is the function f_i(tau)?

3. Mar 29, 2013

6Stang7

fi(τ)=hT

I did state that, but it was after the final solution I got (my mistake for stating it so late).

4. Mar 29, 2013

SteamKing

Staff Emeritus
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. Mar 30, 2013

6Stang7

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. Apr 20, 2013

bynightmare

Matlab project

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