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Math Help - 2D heat equation

  1. #1
    Junior Member raheel88's Avatar
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    2D heat equation

    hello all.

    does anyone know the general solution to the 2d heat equation, namely

    \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}

    subject to

    u(x_1,x_2,\tau = 0) = u_0(x_1,x_2)

    if anyone could help me I'd be well chuffed!
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  2. #2
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    Quote Originally Posted by raheel88 View Post
    hello all.

    does anyone know the general solution to the 2d heat equation, namely

    \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}

    subject to

    u(x_1,x_2,\tau = 0) = u_0(x_1,x_2)

    if anyone could help me I'd be well chuffed!
    Taken directly from David Colton's book - if u_0 is continuous and bounded in absolute then the solution is

     <br />
u(x_1,x_2,\tau) = \frac{1}{4 \pi \tau} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u_0({\xi}_1,{\xi}_2)\, \text{exp} \left[ - \frac{(x_1-{\xi}_1)^2+(x_2-{\xi}_2)^2}{4 \tau} \right] d {\xi}_1 d {\xi}_2<br />
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  3. #3
    Junior Member raheel88's Avatar
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    many thanks danny

    exactly what i was looking for!
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  4. #4
    Junior Member raheel88's Avatar
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    Quote Originally Posted by Danny View Post
    Taken directly from David Colton's book - if u_0 is continuous and bounded in absolute then the solution is

     <br />
u(x_1,x_2,\tau) = \frac{1}{4 \pi \tau} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} u_0({\xi}_1,{\xi}_2)\, \text{exp} \left[ - \frac{(x_1-{\xi}_1)^2+(x_2-{\xi}_2)^2}{4 \tau} \right] d {\xi}_1 d {\xi}_2<br />
    by the way danny, exactly which of his books is this taken from?
    need a reference you see...thanks again!
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  5. #5
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    Quote Originally Posted by raheel88 View Post
    by the way danny, exactly which of his books is this taken from?
    need a reference you see...thanks again!
    Yes, sorry, "Partial Differential Equations - An Introduction."
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