# Heat Conduction in a Rod

## 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$$\pi$$x)-sin(5$$\pi$$x), 0$$\leq$$x$$\leq$$1

## Homework Equations

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

(sum from n=1 to infinity)

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

## The Attempt at a Solution

cn=2$$\int$$(sin(2$$\pi$$x)-sin(5$$\pi$$x))sin(n$$\pi$$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?

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
None of those pi's should be superscripts. Not sure why they came out like that.

HallsofIvy
Homework Helper
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 "$\alpha$" come from? There is no $\alpha$ in your problem. I get
$$e^{-100n^2\pi^2 t}$$
for the exponential.

You don't really need to do any integral.

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