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Math Help - Partial Differential Equations

  1. #1
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    Partial Differential Equations

    Please see attachment for question
    Attached Thumbnails Attached Thumbnails Partial Differential Equations-partialdi.jpg  
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  2. #2
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    Quote Originally Posted by sssouljah View Post
    Please see attachment for question
    This is a nice exercise on the Fourier series solution of PDE. Do you know this theory because it is absolutely necessary if you want to find the solution. If so, can you explain where exactly you are stuck?

    Coomast
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  3. #3
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    Quote Originally Posted by sssouljah View Post
    Please see attachment for question
    <br />
u(x,t)=X(x)\;T(t)<br />

    <br />
u_t=XT_t,\ \ \ \ u_{xx}=TX_{xx}, \ \ \ \ u_t-Du_{xx}-\alpha u=0<br />

    <br />
\frac{T_t}{T}-D\frac{X_{xx}}{X}-\alpha = 0<br />

    <br />
D\frac{X_{xx}}{X}=\frac{T_t}{T}-\alpha=-C\text{ (constant)}<br />

    <br />
u(x,t)=e^{(\alpha-C)t}\left[A\cos\left(\sqrt{\frac{C}{D}}\;x\right)+B\sin\left (\sqrt{\frac{C}{D}}\;x\right)\right]<br />

    <br />
u_x(0,t)=0 \Rightarrow B=0<br />

    <br />
u_x(L,t)=0 \Rightarrow \sin\left(\sqrt{\frac{C}{D}}\;L\right) = 0 \Rightarrow C_n=\frac{n^2\pi^2D}{L^2}, \ n=1, 2, 3, ...<br />

    <br />
u(x,0)=\cos\left(\frac{\pi x}{L}\right)+\cos\left(\frac{2\pi x}{L}\right) \Rightarrow n=1, 2, \ A_1=A_2 =1 <br />

    - <br />
\boxed{u(x,t)=e^{\left(\alpha-\tfrac{\pi^2D}{L^2}\right)t}\cos\left(\frac{\pi x}{L}\;\right)+e^{\left(\alpha-\tfrac{4\pi^2}{L^2}\right)t}\cos\left(\frac{2\pi x}{L}\;\right)}<br />
    Last edited by mr fantastic; September 18th 2009 at 09:21 AM. Reason: Restored original reply
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