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Thread: laplace equation

  1. #1
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    laplace equation

    $\displaystyle \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0, u=-K \quad\mbox{at} \quad x=\pm b, y=\pm h$

    How to solve this?
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  2. #2
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    If you let $\displaystyle u = v - K$ then $\displaystyle v $ satisfies

    $\displaystyle \dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} = 0$

    where $\displaystyle v = 0$ on the boundary of the rectangle. The solution of this second problem is $\displaystyle v \equiv 0$. Thus, the solution of your problem us $\displaystyle u = -K$ on the entire region.
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