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Thread: 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 $\displaystyle u_{t} = u_{xx}$ with initial condition $\displaystyle u(x,0) = f(x)$ can be written as

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

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

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


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

    The next bit is the problem:

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

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

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

    to

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


    This is where I'm stuck. I'm guessing it's some kind of delta function property but I'm not sure. The $\displaystyle \delta^{\left(n\right)}$ 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 $\displaystyle u_{t} = u_{xx}$ with initial condition $\displaystyle u(x,0) = f(x)$ can be written as

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

    [snip]

    The next bit is the problem:

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

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

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

    to

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


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

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

    More generally:

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