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Math Help - Heat Equation

  1. #1
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    Heat Equation

    Could someone help me with the following problem:

    Find all smooth functions u: R x [0, +infinity) ->R with the property that both u and u^n are solutions of the heat equation for some integer n>=2.
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  2. #2
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    Quote Originally Posted by ruggles View Post
    Could someone help me with the following problem:

    Find all smooth functions u: R x [0, +infinity) ->R with the property that both u and u^n are solutions of the heat equation for some integer n>=2.
    I do not know if this will help, but It may give you somewhere to start.

    u_t(x,t)=ku_{xx}(x,t)

    let u(x,t) \mbox{ and } [u(x,t)]^n be solutions to the heat equation

    let z=[u(x,t)]^n

    z_t=nu^{n-1}u_t

    z_x=nu^{n-1}u_x

    z_{xx}=n(n-1)u^{n-2}(u_x)^2+nu^{n-1}u_{xx}

    plugging the above into the heat eqaution gives

    z_t=kz_{xx}

    nu^{n-1}u_t=k[n(n-1)u^{n-2}(u_x)^2+nu^{n-1}u_{xx}]

    nu^{n-1}u_t=kn(n-1)u^{n-2}(u_x)^2+knu^{n-1}u_{xx}

    nu^{n-1}u_t-knu^{n-1}u_{xx}=kn(n-1)u^{n-2}(u_x)^2

    nu^{n-1}\underbrace{[u_t-ku_{xx}]}_{=0 \mbox{ because }u_t=ku_{xx}}=kn(n-1)u^{n-2}(u_x)^2

    0=kn(n-1)u^{n-2}(u_x)^2


    Now if u(x,t)=X(x) \cdot T(t) we get

    0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n-2}[X_x(x)T(t)]^2

    0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n}[X_x(x)]^2

    0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n}[\frac{dX}{dx}]^2

    Maybe you can figure something out from here.

    I hope this helps.

    Good luck.
    Last edited by TheEmptySet; May 20th 2008 at 12:57 PM.
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