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Homework Help: 1D Heat Equation BC: Both ends insulated, IC: piecewise function

  1. Oct 26, 2008 #1

    PAR

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    1. Solve the one dimensional heat equation for a rod of length 1 with the following boundary and initial conditions:

    BC: [tex]\partial[/tex]u(0,t)/[tex]\partial[/tex]t = 0
    [tex]\partial[/tex]u(1,t)/[tex]\partial[/tex]t = 0 these are the wrong boundary conditions (see below)


    Actual BC: [tex]\partial[/tex]u(0,t)/[tex]\partial[/tex]x = 0
    [tex]\partial[/tex]u(1,t)/[tex]\partial[/tex]x = 0


    IC:
    u(x,0) = { 1 if 0[tex]\leq[/tex]x[tex]\leq[/tex].5
    u(x,0) = { 0 if .5<x[tex]\leq[/tex]1




    2. [tex]\partial[/tex]u/[tex]\partial[/tex]t = a2[tex]\partial[/tex][tex]^{2}[/tex]u/[tex]\partial[/tex]x[tex]^{2}[/tex]



    3.

    I used separation of variables and applied the boundary conditions to get the following:
    u(x,t) = e(-(a2)(n*Pi)2*t) * (B*cos(n*Pi*x)) where
    n= 1, 2, 3... and B is an unknown constant.

    To find B I tried applying the initial conditions and thats where I got stuck because I got.

    u(x,0) = 0 = B*cos(n*Pi*x) if 0[tex]\leq[/tex]x[tex]\leq[/tex].5
    u(x,0) = 1 = B*cos(n*Pi*x) if .5<x[tex]\leq[/tex]1

    Does this mean that B has two values depending on x? And if is so are those values:

    B = 0 for 0[tex]\leq[/tex]x[tex]\leq[/tex].5
    B = 1/cos(n*Pi*x) for .5<x[tex]\leq[/tex]1

    As a follow-up, will the steady state of this heat equation problem be a piecewise function?
     
    Last edited: Oct 26, 2008
  2. jcsd
  3. Oct 26, 2008 #2

    gabbagabbahey

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    Shouldn't it be a sine instead of a cosine? (take a look at your BCs cos(0)=1 not 0)

    Also, since that solution satisfies the heat equation and your boundary conditions for any integer value of n, shouldn't the general solution be a linear combination of each of your solutions, namely:

    [itex]u(x,t)=\sum_{n=0}^{\infty} u_n(x,t)=\sum_{n=0}^{\infty}B_ne^{-a^2n^2 \pi^2 t}sin(n \pi x)[/itex]

    ???

    Remember, your B_n's are constants so they cannot depend on x. Use Fourier's method to determine them.
     
  4. Oct 26, 2008 #3

    PAR

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    Sorry, I miss-typed the boundary conditions, the boundary conditions are supposed to be:

    [tex]\partial[/tex]u(0,t)/[tex]\partial[/tex]x = 0
    [tex]\partial[/tex]u(1,t)/[tex]\partial[/tex]x = 0

    I'll edit it to the correct version in the original post too.

    So I think that the solution is the linear combination:

    [tex]u_{n}(x,t) = [/tex][tex]\sum(B_{n}e^{-a^2n^2\pi^2t}cos(n\pi\\x))[/tex] going from n = 1 to infinity

    So now I set the sum equal to initial condition right?

    I get:

    [tex]u_{n}(x,0) = 1 = \sum(B_{n}cos(n\pi\\x))[/tex] for 0<=x<=1/2

    I use fouriers method to find the B's in each interval:

    [tex]B_{n} = 2 \int^{1/2}_{0}cos(n\pi\\x)= 2 sin(n\pi\\x)/(n\pi)\right|^{1/2}_{0}
    = 2(sin(n\pi/2)/(n\pi) = 2(-1)^n/(n\pi)[/tex]

    Unfortunately I'm stuck again, when I try to solve for the B's in the interval 1/2<x<=1

    I get:

    [tex]u_{n}(x,0) = 0 = \sum(B_{n}cos(n\pi\\x))[/tex]

    [tex]B_{n} = 2 \int^{1}_{1/2} 0*cos(n\pi\\x) = 0 [/tex]

    Is this correct? Are the B's for 1/2<x<=1 all 0?

    And sorry again about giving the wrong boundary conditions!
     
  5. Oct 26, 2008 #4

    gabbagabbahey

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    Okay, so doesn't that mean that

    [itex] \sum_{n=0}^{\infty} B_n cos(n \pi x)=u(x,0)=\left\{ \begin{array}{lr} 1, & 0 \leq x \leq 0.5 \\ 0, & 0.5< x \leq 1 \end{array}[/itex]

    [itex]\Rightarrow \sum_{m=0}^{\infty} \int_0^1 B_m cos(m \pi x) cos(n \pi x)dx = \int_0^1 u(x,0) cos(n \pi x) dx=\int_0^{0.5} (1)cos(n \pi x) dx+ \int_{0.5}^1 (0)cos(n \pi x) dx[/itex]

    ;0)

    Remember, your B_n's are constant so they must be the same for all values of x.
     
  6. Oct 26, 2008 #5

    PAR

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    Ok,

    so [tex]\int^{1}_{0}cos(n\pi\\x)*cos(m\pi\\x) = 0[/tex] if [tex] m \neq n[/tex]

    so, I've taken the integral of

    [tex]\int^{1}_{0}cos(n\pi\\x)^2dx[/tex]

    And assuming n = positive integer I get

    [tex]\int^{1}_{0}cos(n\pi\\x)^2dx = 1/2 [/tex]

    and I know that [tex]\int^{1}_{0}u(x,0)cos(n\pi\\x)dx = -1^n/(n\pi) [/tex]

    So I get

    [tex]\sum^{\infty}_{n=1}B_{n}(1/2) = (-1)^n/(n\pi) [/tex]

    Reading from my textbook and online I am guessing I should multiply both sides by 2 (or divide by 1/2) which would yield the same answer as in my previous post:

    [tex]B_{n} = 2(-1)^n/(n\pi)[/tex]

    But I am confused by this last step: How does multiplying by both sides by 2 "remove" the [tex]B_{n}[/tex] from out of the summation? Wouldn't it just remove the 2 from the summation and the left hand side of the equation would just be

    [tex]\sum^{\infty}_{n=1}B_{n}[/tex]?

    I realize that I am expected to know these things before I enter a partial differential equations course, but after taking 7 college math classes I still haven't learned these basic principles of sums and it's really holding me back.

    Oh, and thanks for the help so far gabbagabbahey, but I still have much to learn.
     
  7. Oct 26, 2008 #6

    gabbagabbahey

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    The summation sign is removed when you integrate [itex]\sum_{m=0}^{\infty} \int_0^1 B_m cos(m \pi x) cos(n \pi x)dx[/itex], because the only term that is non-zero is m=n

    [itex]\Rightarrow \sum_{m=0}^{\infty} \int_0^1 B_m cos(m \pi x) cos(n \pi x)dx=\frac{B_n}{2}+ \sum_{m \neq n} (0)B_m=\frac{B_n}{2}[/itex]

    [itex]\Rightarrow \frac{B_n}{2}=\frac{(-1)^n}{n \pi}[/itex]

    or

    [itex]B_n=\frac{2(-1)^n}{n \pi}[/itex]
     
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