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Math Help - Need help with PDE problem

  1. #1
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    Need help with PDE problem

    Could anyone help me with this? The question is under the spoiler tag.

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    The PDE equation is separable, but the boundary conditions are non-homogeneous and are in partial derivatives.

    I can answer if the equations are all homogeneous or if the boundary conditions are not homogeneous and not derivatives. But I'm not quite sure how to proceed with this one.

    I would appreciate any help.
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  2. #2
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    Quote Originally Posted by Mierin View Post
    Could anyone help me with this? The question is under the spoiler tag.

    Spoiler:



    The PDE equation is separable, but the boundary conditions are non-homogeneous and are in partial derivatives.

    I can answer if the equations are all homogeneous or if the boundary conditions are not homogeneous and not derivatives. But I'm not quite sure how to proceed with this one.

    I would appreciate any help.

    It looks like a Diffusion Equation..
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  3. #3
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    Quote Originally Posted by Mierin View Post
    Could anyone help me with this? The question is under the spoiler tag.

    Spoiler:



    The PDE equation is separable, but the boundary conditions are non-homogeneous and are in partial derivatives.

    I can answer if the equations are all homogeneous or if the boundary conditions are not homogeneous and not derivatives. But I'm not quite sure how to proceed with this one.

    I would appreciate any help.
    If you let u = a(x^2+2t) + bx + v, then you can choose a and b such that you BC's becomes v_x = 0 at both boundaries.
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  4. #4
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    A function that is equal to -1 at x= 0 and 0 at x= l is \frac{x}{l}- 1. Integrating that, a function whose derivative is -1 at x= 0 and 0 at x= l is \frac{x^2}{2l}- x.

    If you let v(x,t)= u(x,t)- \frac{x^2}{2l}+ x, then \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial u^2} - \frac{1}{l} = \frac{1}{\kappa^2}\frac{\partial u}{\partial t}= \frac{1}{\kappa^2}\frac{\partial v}{\partial t}.

    That is, you now have the differential equation
    \frac{1}{\kappa^2}\frac{\partial v}{\partial t}= \frac{\partial^2 v}{\partial x^2}- \frac{1}{l}

    with boundary conditions
    \frac{\partial v}{\partial x}(0)= \frac{\partial u}{\partial x}(0)- \frac{(0}{l}+ 1= -1+ 1= 0

    and initial condition v(x, 0)= u(x, 0)- \frac{x^2}{2l}+ x= -\frac{x^2}{2l}+ x

    With those boundary conditions, you can write v as a Fourier cosine series:

    v(x,t)= \sum_{n=0}^\infty C_n(t)sin(\frac{2\pi}{l}x).
    \frac{\partial v}{\partial x}(l)= \frac{\partial u}{\partial x}(l)- \frac{(0}{l}+ 1= 0 -1+ 1= 0
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  5. #5
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    Thanks a lot for your help. But I still can't seem to understand how this part: \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial u^2} - \frac{1}{l} = \frac{1}{\kappa^2}\frac{\partial u}{\partial t}= \frac{1}{\kappa^2}\frac{\partial v}{\partial t} came about.


    I can see that \frac{1}{\kappa^2}\frac{\partial u}{\partial t} is obtained from the equation given in the question, but I don't understand \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial u^2} - \frac{1}{l}
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  6. #6
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    Quote Originally Posted by Mierin View Post
    Thanks a lot for your help. But I still can't seem to understand how this part: \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial u^2} - \frac{1}{l} = \frac{1}{\kappa^2}\frac{\partial u}{\partial t}= \frac{1}{\kappa^2}\frac{\partial v}{\partial t} came about.


    I can see that \frac{1}{\kappa^2}\frac{\partial u}{\partial t} is obtained from the equation given in the question, but I don't understand \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial u^2} - \frac{1}{l}
    It's a typo - I think he meant

    \frac{\partial^2 v}{\partial x^2}= \frac{\partial^2 u}{\partial x^2} - \frac{1}{l}

    although if you follow my idea your PDE will be the same and hence the usual separation of variables will work.
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