What is the Solution to a Heat Conduction Problem in a Rod?

[EV33]

Homework Statement

Find the solution of the heat conduction problem:

100U_xx=U_t, 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)=$$\sum$$c_ne^{(-n2$$\pi$$2$$\alpha$$2t)/L2}sin((n$$\pi$$x)/L)

(sum from n=1 to infinity)


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

The Attempt at a Solution

c_n=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 c_n then what should I do different?


Thank you for your time.

 

[EV33]

None of those pi's should be superscripts. Not sure why they came out like that.

 

[HallsofIvy]

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.