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

$$X:=C_1\cos kx+C_2\sin kx$$

and for  $Y^{\prime \prime }=k^{2}Y$  the solution

$$Y:=D_1\cosh ky+D_2\sinh ky.$$

The two first boundary conditions give  $0=X(0)=C_1$,  $0=X(\pi )=0+C_2\sin k\pi$,  and since  $C_2\ne 0$,  we must have  $\sin k\pi =0$,  i.e.

Therefore

$$X(x):=C_2\sin nx,Y^{\prime }(y)\equiv n{D}_1\sinh ny+n{D}_2\cosh ny.$$

The fourth boundary condition now yields that  $0=Y^{\prime }(0)=n{D}_2$; thus  $D_2=0$  and  $Y(y):=D_1\cosh ny.$  So (1) has infinitely many solutions

$u_n:=C_2{D}_1\sin nx\cosh ny=A_n\sin nx\cosh ny$(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

$$u:=\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin nx\cosh ny$$

of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges.  The third boundary condition requires that

$$C=u(x,\pi )=\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin nx\cosh n\pi =\sum _{n=1}^{\mathrm{\infty }}(A_n\cosh n\pi )\sinh nx$$

on the interval  $0\leqq x\leqq \pi$.  But this is the Fourier sine series of the constant function  $x\mapsto C$  on the half-interval  $[0,\pi ]$,  whence

$$A_n\cosh n\pi =\frac{2}{\pi }{\int }_0^{\pi }C\sin nxdx=\frac{2C}{n\pi }(1-{(-1)}^n)\mathit{\hspace{1em}}\forall n\in {\mathbb{Z}}_{+}.$$

The even (http://planetmath.org/EvenNumber) $n$’s here give 0 and the odd (http://planetmath.org/EvenNumber) give

$$A_{2m+1}:=\frac{4C}{(2m+1)\pi \cosh (2m+1)\pi }\mathit{\hspace{1em}}(m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$$

Thus we obtain the solution

$$u(x,y):=\frac{4C}{\pi }\sum _{m=0}^{\mathrm{\infty }}\frac{\sin (2m+1)x\cosh (2m+1)y}{(2m+1)\cosh (2m+1)\pi }.$$

It can be shown that this series converges in the whole of the plate.

Visualization of the solution

Figure 1: Surface plot of the solution $u(x,y)$, for $C=1$

Figure 2: Color-coded plot of the temperature $u(x,y)$

Remark.  The function $u$ has been approximated in the plot by computing a partial sum of the true infinite-series 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

$$\frac{4C}{\pi }\left(\sin x+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\mathrm{\cdots }\right)$$

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.

• http://gold-saucer.afraid.org/math/planetmath/ExampleOfSolvingTheHeatEquation/heat.pyPython program to compute  $u(x,y)$  and produce the two figures