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Math Help - Using fourier and laplace transform to solve PDE

  1. #1
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    Using fourier and laplace transform to solve PDE

    PDE is type of heat equation.
    Where have above reference notes?


    Many book only gives an example of solving heat equation using fourier transform.
    An exercise asks me to solve it for using fourier and laplace transform:
    u_xx = u_t -inf < x < +inf, t >0
    u(x,o) = x
    u(o,t) = 0


    In the heat equation, we'd take the fourier transform with respect to x for
    each term in the equation. How to combine it with using fourier and laplace transform



    Can anyone suggest some example and notes to me??
    Last edited by cyw1984; August 5th 2007 at 08:23 AM.
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  2. #2
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    Can anyone provide some example about this?
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  3. #3
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    Quote Originally Posted by cyw1984 View Post
    An exercise asks me to solve it for using fourier and laplace transform:
    u_xx + u_t = 0 -inf < x < +inf, t >0
    The heat equation is,
    u_{t}=c^2u_{xx}
    Now the way you have it is,
    u_{t} = -u_{xx}
    How is that possible?
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  4. #4
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    O...Sorry...correct it>_<
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  5. #5
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    And you forgot to mention the initial value problem.
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  6. #6
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    Quote Originally Posted by ThePerfectHacker View Post
    And you forgot to mention the initial value problem.
    yES///i Want to have some example^^ But I can not find related this >_<
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  7. #7
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    Quote Originally Posted by cyw1984 View Post
    An exercise asks me to solve it for using fourier and laplace transform:
    u_xx = u_t -inf < x < +inf, t >0
    u(x,o) = x
    u(o,t) = 0
    The condition u(0,t)=0 are only used in finite regions when a boundary value problem exists.

    A valid problem would be,
    u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x

    The solution is given by,
    u(x,t) = \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}f(x+2y\sqrt{t})e^{-y^2}dy= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}(x+2y\sqrt{t})e^{-y^2} dy
    Is the solution.
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  8. #8
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    Quote Originally Posted by ThePerfectHacker View Post
    The condition u(0,t)=0 are only used in finite regions when a boundary value problem exists.

    A valid problem would be,
    u_{xx} = u_t \mbox{ for } -\infty< x < \infty \mbox{ and } t>0 \mbox{ with }u(x,0)=f(x)=x

    The solution is given by,
    u(x,t) = \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}f(x+2y\sqrt{t})e^{-y^2}dy= \frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}(x+2y\sqrt{t})e^{-y^2} dy
    Is the solution.
    umum.....the question mention us to use fourier and laplace transform. Is there mention us to use fourier transform with respect to t (t>0) and then inverse fourier transform to solve the solution; Separately, solve the question using laplace transform with respect to x -\infty< x < \infty and then inverse laplace transform?

    Can there combines two transformation in one??
    What is the result compared with two method??

    Thanks
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