1. ## boundary value problem

Hi,

I'm taking up a course in partial diff equations and i'm really lost here. I cant seem to find the solution for the problem that the professor discussed.

here it is:

phi(x) = sin(2*pi*x) + (1/3)*sin(4*pi*x) + (1/5)*sin(6*pi*x)

b.c.:

u(0,t) = 0
u(1,t) = 0

0<t<infinity

can anyone help?

2. Originally Posted by mousemouse
Hi,

I'm taking up a course in partial diff equations and i'm really lost here. I cant seem to find the solution for the problem that the professor discussed.

here it is:

phi(x) = sin(2*pi*x) + (1/3)*sin(4*pi*x) + (1/5)*sin(6*pi*x)

b.c.:

u(0,t) = 0
u(1,t) = 0

0<t<infinity

can anyone help?
Where is the differencial equation?

I am assuming you are solving $\displaystyle \frac{\partial u}{\partial t} = k^2 \frac{\partial ^2 u}{\partial x^2}$ given the initial value problem $\displaystyle u(x,0) = \phi (x)$?

3. ## sorry,

I seem to have copied incomplete notes, i cant seem to find it anywhere. Anyways, this is another problem which i dont understand.

mu(t) = (alpha^2)(Uxx)

0<x<1
0<t<infinity

b.c.

u(0,t) = 0
u(1,t) = 0

i.c.

u(x,0) = x
while 0<x<1

P.S.: if the "mu" doesnt seem right to you, change it to just "u". i cant seem to understand my bad handwriting lol.

Also, I was reading partial differential equations by farlow, and it seemed ok to give a background but doesnt give detailed examples for noobs like me. Any reading suggestions?

thanks

4. ## My suggestion for reading

would be anything by Schaum's outline or anything by Dover I love them...

5. Originally Posted by mousemouse
u_t = (alpha^2)(Uxx)

0<x<1
0<t<infinity

b.c.

u(0,t) = 0
u(1,t) = 0

i.c.

u(x,0) = x
while 0<x<1
The solution to this equation is given by,
$\displaystyle u(x,t) = \sum_{n=1}^{\infty} A_n \sin (\pi n x ) e^{-n^2 \pi^2 \alpha^2 t}$

Where $\displaystyle A_n = 2\int_0^1 x\sin (\pi nx) dx$

6. ## thanks

Originally Posted by ThePerfectHacker
The solution to this equation is given by,
$\displaystyle u(x,t) = \sum_{n=1}^{\infty} A_n \sin (\pi n x ) e^{-n^2 \pi^2 \alpha^2 t}$

Where $\displaystyle A_n = 2\int_0^1 x\sin (\pi nx) dx$
thanks

7. Originally Posted by mousemouse
thanks
You know what to do from there? You should evaluate that integral and you will get an expression for $\displaystyle n$, that would be the terms of the sequence $\displaystyle A_n$ which go into that infinite sum.

8. ## Just to help

$\displaystyle \int{xsin(x\pi{n})}dx=\frac{-xcos(x\pi{n})}{n\pi}+\frac{sin(x\pi{n})}{n^2{\pi}^ 2}+C$

9. Originally Posted by ThePerfectHacker
You know what to do from there? You should evaluate that integral and you will get an expression for $\displaystyle n$, that would be the terms of the sequence $\displaystyle A_n$ which go into that infinite sum.
I "think" i do. lol. I have some examples and they are pretty strightforward once i get the solutions. I'd try doing this on my own as i think it is manageable for me

anyway,

do u mind if i ask you for another solution? I try reading the flow of proofs online, but i cant do anything on my own, save for the simplest ones.

here it is:

u(t) = Uxx - U
0<x<1
0<t<infinity

b.c.
u(0,t) = 0
u(1,t) = 0

i.c.

u(x,0) = sin(pi*x) + 0.5*sin(3*pi**x)
when 0<x<1

10. ## bump

can anyone help with the problem?