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Math Help - Non-Homogenous Heat Equation

  1. #1
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    Non-Homogenous Heat Equation

    Consider the following non-homogenous heat equation on 0 \leq x \leq \pi

    u_t = k u_{xx} - 1 with u(x,0) = 0, u(0,t) = 0, u(\pi, t) = 0

    Find a solution of the form

    \displaystyle \sum_1^{\infty} b_n(t) \phi_n (x)

    where \phi_n(x) are the eigenfunctions of an appropriate homogenous problem, and find explicit expressions for b_n(t)
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  2. #2
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    So I think

    \phi_n(x) = \sin \frac{n \pi x}{L}

    so I find solutions in the form

    \displaystyle u(x,t) = \sum_1^{\infty} b_n (t) \ \sin \frac{n \pi x}{L}

    Am I on the right track? Is the eigenfunction correct?
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  3. #3
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    You are on the right track. You'll need a Fourier sine series for -1 and then sub in your expansion and equate coefficients.
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  4. #4
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    Quote Originally Posted by JoernE View Post
    So I think

    \phi_n(x) = \sin \frac{n \pi x}{L}

    so I find solutions in the form

    \displaystyle u(x,t) = \sum_1^{\infty} b_n (t) \ \sin \frac{n \pi x}{L}

    Am I on the right track? Is the eigenfunction correct?
    Okay, ignore this last post. I have a different method....

    Let u = w(x,t) + v(x)

    Then

    \displaystyle \frac{\partial u}{\partial t} = \frac{\partial w}{\partial t} and \displaystyle \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 v}{\partial x^2}

    Sub these back into the PDE to obtain

    \displaystyle \frac{\partial w}{\partial t} = k \frac{\partial^2 w}{\partial x^2} + k \frac{\partial^2 v}{\partial x^2}  - 1

    Choose \displaystyle k \frac{\partial^2 v}{\partial x^2} - 1 = 0

    So

    \displaystyle V'' = \frac{1}{k}, \ V(0) = V(\pi) = 0

    \displaystyle V' = \frac{1}{k} + A

    \displaystyle V = \frac{1}{k} + Ax + B


    \displaystyle V(0) = \frac{1}{k} + B = 0 \ \Rightarrow \ B = - \frac{1}{k}

    \displaystyle V(\pi) = \frac{1}{k} + A \pi + B \ \Rightarrow \ A = -\frac{1}{k \pi} - \frac{B}{\pi}

    \displaystyle V(x) = 0??

    Also since \displaystyle k \frac{\partial^2 v}{\partial x^2} - 1 = 0 we have

    \displaystyle \frac{\partial w}{\partial t} = k \frac{\partial^2 w}{\partial x^2} which is homogeneous.

    \displaystyle u(0, t) = v(0)  + w(0, t) = 0

    \displaystyle \Rightarrow \ w(0,t) = 0

    \displaystyle u(x, 0) = v(x) + w(x,0) = 0

    \displaystyle \Rightarrow \ w(x,0) = 0

    \displaystyle u(\pi, t) = v(\pi) + w(\pi, t) = 0

    \displaystyle \Rightarrow \ w(\pi, t) = 0

    Then we have

    \displaystyle w_t = k w_{xx} with \displaystyle w(x,o) = 0, w(0,t) = 0, w(\pi, t) = 0.

    Since this is homogenous, we can solve by letting

    \displaystyle w = XT

    \displaystyle \frac{T'}{kT} = \frac{X''}{X} = - \lambda

    \displaystyle X = A \cos(\sqrt{\lambda}t) + B \sin(\sqrt{\lambda}t)

    \displaystyle X(0) = A = 0

    but now I get a little lost.....I think I know where to go, just not exactly how to get there. Already have calculated that v(x) = 0, which will lead to what I need to prove (I think).
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  5. #5
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    Quote Originally Posted by JoernE View Post

    \displaystyle X = A \cos(\sqrt{\lambda}t) + B \sin(\sqrt{\lambda}t)
    This should be \displaystyle X = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)

    so

    \displaystyle X(\pi) = B \sin (\sqrt{\lambda} \pi) = 0

    \displaystyle \ \sqrt{\lambda} \pi = \frac{n \pi }{2}

    \displaystyle \ \sqrt{\lambda} = \frac{n}{2}

    \displaystyle \ \lambda_n = \big( \frac{n}{2} \big)^2 for n = 1,2,...

    So

    \displaystyle \phi_n(x) = \sin \big(\frac{n \pi x}{2} \big)

    and so

    \displaystyle u(x,t) = v(x) + \sum_1^{\infty} b_n \sin \big(\frac{n \pi}{2} \big)

    \displaystyle \ = \displaystyle \sum_1^{\infty} b_n(t) \phi_n (x)<br />

    and I think

    \displaystyle \ b_n = \frac{1}{\pi} \int_0^{\pi} \sin(\sqrt{\lambda} x) dx

    Could someone please tell me if this is now correct?

    Thanks
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