example of solving the heat equation
Let a . Determine the temperature function $(x,y)\mapsto u(x,y)$ on the plate, when the faces of the plate are .
The equation of the heat flow (http://planetmath.org/HeatEquation) in this case is
${\nabla}^{2}u\equiv {u}_{xx}^{\prime \prime}+{u}_{yy}^{\prime \prime}=\mathrm{\hspace{0.33em}0}$  (1) 
under the boundary conditions^{}
$$u(0,y)=0,u(\pi ,y)=0,u(x,\pi )=C,{u}_{y}^{\prime}(x,\mathrm{\hspace{0.17em}0})=0.$$ 
We first try to separate the variables, i.e. seek the solution of (1) of the form
$$u(x,y):=X(x)Y(y).$$ 
Then we get
$${u}_{x}^{\prime}={X}^{\prime}Y,{u}_{xx}^{\prime \prime}={X}^{\prime \prime}Y,{u}_{y}^{\prime}=X{Y}^{\prime},{u}_{yy}^{\prime \prime}=X{Y}^{\prime \prime},$$ 
and thus (1) gets the form
${X}^{\prime \prime}Y+X{Y}^{\prime \prime}=\mathrm{\hspace{0.33em}0}$  (2) 
and the boundary conditions
$$X(0)=X(\pi )=\mathrm{\hspace{0.33em}0},X(x)=\frac{C}{Y(\pi )},{Y}^{\prime}(0)=\mathrm{\hspace{0.33em}0}.$$ 
We separate the variables in (2):
$$\frac{{X}^{\prime \prime}}{X}=\frac{{Y}^{\prime \prime}}{Y}$$ 
This equation is not possible unless both sides are equal to a same negative ${k}^{2}$, which implies for ${X}^{\prime \prime}={k}^{2}X$ the solution
$$X:={C}_{1}\mathrm{cos}kx+{C}_{2}\mathrm{sin}kx$$ 
and for ${Y}^{\prime \prime}={k}^{2}Y$ the solution
$$Y:={D}_{1}\mathrm{cosh}ky+{D}_{2}\mathrm{sinh}ky.$$ 
The two first boundary conditions give $0=X(0)={C}_{1}$, $0=X(\pi )=0+{C}_{2}\mathrm{sin}k\pi $, and since ${C}_{2}\ne 0$, we must have $\mathrm{sin}k\pi =0$, i.e.
$$ 
Therefore
$$X(x):={C}_{2}\mathrm{sin}nx,{Y}^{\prime}(y)\equiv n{D}_{1}\mathrm{sinh}ny+n{D}_{2}\mathrm{cosh}ny.$$ 
The fourth boundary condition now yields that $0={Y}^{\prime}(0)=n{D}_{2}$; thus ${D}_{2}=0$ and $Y(y):={D}_{1}\mathrm{cosh}ny.$ So (1) has infinitely many solutions
${u}_{n}:={C}_{2}{D}_{1}\mathrm{sin}nx\mathrm{cosh}ny={A}_{n}\mathrm{sin}nx\mathrm{cosh}ny$  (3) 
with $n\in {\mathbb{Z}}_{+}$ and they all satisfy the boundary conditions except the third. Because of the linearity of (1), also the sum
$$u:=\sum _{n=1}^{\mathrm{\infty}}{A}_{n}\mathrm{sin}nx\mathrm{cosh}ny$$ 
of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges. The third boundary condition requires that
$$C=u(x,\pi )=\sum _{n=1}^{\mathrm{\infty}}{A}_{n}\mathrm{sin}nx\mathrm{cosh}n\pi =\sum _{n=1}^{\mathrm{\infty}}({A}_{n}\mathrm{cosh}n\pi )\mathrm{sinh}nx$$ 
on the interval $0\leqq x\leqq \pi $. But this is the Fourier sine series^{} of the constant function $x\mapsto C$ on the halfinterval $[0,\pi ]$, whence
$${A}_{n}\mathrm{cosh}n\pi =\frac{2}{\pi}{\int}_{0}^{\pi}C\mathrm{sin}nxdx=\frac{2C}{n\pi}(1{(1)}^{n})\mathit{\hspace{1em}}\forall n\in {\mathbb{Z}}_{+}.$$ 
The even (http://planetmath.org/EvenNumber) $n$’s here give 0 and the odd (http://planetmath.org/EvenNumber) give
$${A}_{2m+1}:=\frac{4C}{(2m+1)\pi \mathrm{cosh}(2m+1)\pi}\mathit{\hspace{1em}}(m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots})$$ 
Thus we obtain the solution
$$u(x,y):=\frac{4C}{\pi}\sum _{m=0}^{\mathrm{\infty}}\frac{\mathrm{sin}(2m+1)x\mathrm{cosh}(2m+1)y}{(2m+1)\mathrm{cosh}(2m+1)\pi}.$$ 
It can be shown that this series converges in the whole of the plate.
Visualization of the solution
Remark. The function $u$ has been approximated in the plot by computing a partial sum of the true infiniteseries solution. However, there is substantial numerical error in the approximate solution near $y=\pi $, evident in the small oscillations observed in the surface plot, that should not be there in . This phenomenon is actually inevitable given that the boundary conditions are actually discontinuous^{} at the corners $(0,\pi )$ and $(\pi ,\pi )$.
More precisely, observe that when $y=\pi $, the for $u(x,y)$ reduces to the Fourier series
$$\frac{4C}{\pi}\left(\mathrm{sin}x+\frac{\mathrm{sin}3x}{3}+\frac{\mathrm{sin}5x}{5}+\mathrm{\cdots}\right)$$ 
for the discontinuous function on $[\pi ,\pi ]$:
$$ 
That means the Fourier will necessarily be subject to the Gibbs phenomenon. Of course, the series also cannot converge absolutely; in other of the series decay too slowly in magnitude, adversely affecting the numerical solution.

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http://goldsaucer.afraid.org/math/planetmath/ExampleOfSolvingTheHeatEquation/heat.pyPython program to compute $u(x,y)$ and produce the two figures
Title  example of solving the heat equation 
Canonical name  ExampleOfSolvingTheHeatEquation 
Date of creation  20140928 17:02:21 
Last modified on  20140928 17:02:21 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  25 
Author  pahio (2872) 
Entry type  Example 
Classification  msc 35Q99 
Synonym  stationary example of heat equation 
Related topic  LaplacesEquation 
Related topic  BlackScholesPDE 
Related topic  AnalyticSolutionOfBlackScholesPDE 
Related topic  SolvingTheWaveEquationByDBernoulli 
Related topic  TimeDependentExampleOfHeatEquation 
Related topic  ExampleOfSummationByParts 