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

Title	example of solving the heat equation
Canonical name	ExampleOfSolvingTheHeatEquation
Date of creation	2014-09-28 17:02:21
Last modified on	2014-09-28 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