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Heat Conduction in a Rod

  • Thread starter EV33
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Homework Statement


Find the solution of the heat conduction problem:

100Uxx=Ut, 0<x<1, t>0;

U(0,t)=0, U(1,t)=0, t>0;

U(x,0)=sin(2[tex]\pi[/tex]x)-sin(5[tex]\pi[/tex]x), 0[tex]\leq[/tex]x[tex]\leq[/tex]1

Homework Equations



U(x,t)=[tex]\sum[/tex]cne(-n2[tex]\pi[/tex]2[tex]\alpha[/tex]2t)/L2sin((n[tex]\pi[/tex]x)/L)

(sum from n=1 to infinity)


cn=2/L [tex]\int[/tex]f(x)sin((n[tex]\pi[/tex]x)/L)dx (evalutated from 0 to L)

The Attempt at a Solution




cn=2[tex]\int[/tex](sin(2[tex]\pi[/tex]x)-sin(5[tex]\pi[/tex]x))sin(n[tex]\pi[/tex]x)dx=0 (evalutated from 0 to 1)

I did this tedious integral by hand and got zero and verified it with my calculator. Therefore, I think that I am setting up the problem wrong. So if this is not how to set up cn then what should I do different?


Thank you for your time.
 
Last edited:

Answers and Replies

  • #2
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None of those pi's should be superscripts. Not sure why they came out like that.
 
  • #3
HallsofIvy
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Strictly speaking they didn't. LaTeX is typically put slightly out of line with the text. I recommend that you NOT put indvidual symbols in LaTeX but entire formulas. Using "[ itex ]... [ /itex ]" (without the spaces, of course) will keep short formulas better in line with text. [ tex ]... [ /tex ] will look better on separate lines from text.

Where did that "[itex]\alpha[/itex]" come from? There is no [itex]\alpha[/itex] in your problem. I get
[tex]e^{-100n^2\pi^2 t}[/tex]
for the exponential.

You don't really need to do any integral.

I get
[tex]\sum C_n e^{-100n^2\pi^2 t}sin(n\pi x)[/tex]
and at t= 0 that is
[tex]\sum C_n sin(n\pi x)= sin(2\pi x)- sin(5\pi x)[/tex]
It should be obvious from that what each [itex]C_n[/itex] must be.
 

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