Suppose the rectangular region $0 < x < 2, 0 < y < 5$ is a thin plate in which the temperature u is a function of time t and position (x,y). If the initial temperature is f(x,y) and the boundaries are held at temperature for $t > 0$ then u satisfies the two-dimensional heat equation $$\displaystyle k\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) =\frac{\partial u}{\partial t},\hskip 15pt 0< x < 2, 0<y<5, t > 0$$ $$\displaystyle u\left(0,y,t\right) = 0,\hskip 5pt u\left(2,y,t\right) = 0,\hskip 35pt 0<y<5, t>0$$ $$\displaystyle u\left(x,0,t\right) = 0,\hskip 5pt u\left(x,5,t\right) = 0,\hskip 35pt 0<x<2, t>0$$ $$\displaystyle u\left(x,y,0\right) = \left(x-2\right)\!\left(y-5\right),\hskip 35pt 0<x<2,0<y<5$$
Then $\displaystyle u\left(x,y,t\right) = \sum\limits_{n=1}^\infty A_{m,n}e^{-k\!\left(\left(\frac{m\pi }{2}\right)^{2}+\left(\frac{n\pi }{5}\right)^{2}\right)t}\sin\!\left(\frac{m\pi x}{2}\right)\sin\!\left(\frac{n\pi y}{5}\right)$
where

$A_{m,n} =$

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$\displaystyle\int\limits_0$

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dxdy

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