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Math Help - PDEs - Heat equation

  1. #1
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    PDEs - Heat equation

    Hi, I have a heat equation question which I'm unsure of. The first two parts are ok, but I'll post them anyway so you have a bit of background info:

    ============================
    a) Show that the solution of the heat equation u_{t} = u_{xx} with initial condition u(x,0) = f(x) can be written as

    u(x,t) = \int_{-\infty}^{\infty} \frac{1}{2\sqrt{{\pi}t}}\exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'

    b) Show that the solutions of the heat equation can be mapped into non-trivial solutions of Burger's equation v_{t} + vv_{x} = v_{xx} by the Cole-Hopf transformation

    v = \frac{{\alpha}u_{x}}{u}
    and find \alpha


    These are just bookwork. We get \alpha = -2 for part b).
    ============================

    The next bit is the problem:

    c) By taking the singular data for the heat equation f(x) = \delta^{\left(n\right)}(x) construct a solution u(x,t) of the heat equation.

    Now, I've gone through the solutions and the line has jumped from:

    u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\int_{-\infty}^{\infty} \exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'

    to

    u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\left(-\frac{\partial}{{\partial}x}\right)^{n}\exp\left(-\frac{x^{2}}{4t}\right)


    This is where I'm stuck. I'm guessing it's some kind of delta function property but I'm not sure. The \delta^{\left(n\right)} threw me a little. If someone can explain what's going on, that'd be great!

    Thanks.
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  2. #2
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    Quote Originally Posted by sathys View Post
    [snip]
    the solution of the heat equation u_{t} = u_{xx} with initial condition u(x,0) = f(x) can be written as

    u(x,t) = \int_{-\infty}^{\infty} \frac{1}{2\sqrt{{\pi}t}}\exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'

    [snip]

    The next bit is the problem:

    c) By taking the singular data for the heat equation f(x) = \delta^{\left(n\right)}(x) construct a solution u(x,t) of the heat equation.

    Now, I've gone through the solutions and the line has jumped from:

    u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\int_{-\infty}^{\infty} \exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'

    to

    u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\left(-\frac{\partial}{{\partial}x}\right)^{n}\exp\left(-\frac{x^{2}}{4t}\right)


    This is where I'm stuck. I'm guessing it's some kind of delta function property but I'm not sure. The \delta^{\left(n\right)} threw me a little. If someone can explain what's going on, that'd be great!

    Thanks.
    \delta^{\left(n\right)}(x) denotes the nth derivative of the Dirac delta function. The 'jump' uses a property of the nth derivative of the Dirac delta function.

    For example, \int_{-\infty}^{+ \infty} \delta^{\left(1\right)}(x') \, f(x', x) \, dx' = -\frac{df(x)}{dx}.

    More generally:

    \int_{-\infty}^{+ \infty} \delta^{\left(n\right)}(x') \, f(x', x) \, dx' = (-1)^n \frac{d^n f(x)}{dx^n}.
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