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Math Help - boundary value problem

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by mousemouse View Post
    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 \frac{\partial u}{\partial t} = k^2 \frac{\partial ^2 u}{\partial x^2} given the initial value problem u(x,0) = \phi (x)?
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  3. #3
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    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
    Last edited by ThePerfectHacker; April 6th 2008 at 07:31 PM.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    My suggestion for reading

    would be anything by Schaum's outline or anything by Dover I love them...
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  5. #5
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    Quote Originally Posted by mousemouse View Post
    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,
    u(x,t) = \sum_{n=1}^{\infty} A_n \sin (\pi n x ) e^{-n^2 \pi^2 \alpha^2 t}

    Where A_n = 2\int_0^1 x\sin (\pi nx) dx
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  6. #6
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    thanks

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

    Where A_n = 2\int_0^1 x\sin (\pi nx) dx
    thanks
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  7. #7
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    Quote Originally Posted by mousemouse View Post
    thanks
    You know what to do from there? You should evaluate that integral and you will get an expression for n, that would be the terms of the sequence A_n which go into that infinite sum.
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  8. #8
    MHF Contributor Mathstud28's Avatar
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    Just to help

    \int{xsin(x\pi{n})}dx=\frac{-xcos(x\pi{n})}{n\pi}+\frac{sin(x\pi{n})}{n^2{\pi}^  2}+C
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  9. #9
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    Quote Originally Posted by ThePerfectHacker View Post
    You know what to do from there? You should evaluate that integral and you will get an expression for n, that would be the terms of the sequence 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
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  10. #10
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    bump

    can anyone help with the problem?
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