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example of solving the heat equation

(Example)

Let a thin square-formed plate of heat conducting homogeneous material be in the -plane with sides on the -axis (isolated), on the line $y = \pi$ (held at the constant temperature ), and on the vertical lines and $x = \pi$ (both held at the constant temperature ). Determine the temperature function $(x, y)\mapsto u(x, y)$ on the plate, when the faces of the plate are isolated.

The equation of the heat flow in this stationary case is

$\displaystyle \nabla^2 u \equiv u”_{xx}+u”_{yy} = 0$

(1)

under the boundary conditions

$\displaystyle u(0, y) = 0, u(\pi, y) = 0, u(x, \pi) = C, u'_y(x, 0) = 0.$

We first try to separate the variables, i.e. seek the solution of (1) of the form

$\displaystyle u(x, y) := X(x)Y(y).$

Then we get

$\displaystyle u'_x = X'Y, u”_{xx} = X”Y, u'_y = XY', u”_{yy} = XY”,$

and thus (1) gets the form

$\displaystyle X”Y+XY” = 0$

(2)

and the boundary conditions

$\displaystyle X(0) = X(\pi) = 0, X(x) = \frac{C}{Y(\pi)}, Y'(0) = 0.$

We separate the variables in (2):

$\displaystyle \frac{X”}{X} = -\frac{Y”}{Y}$

This equation is not possible unless both sides are equal to a same negative constant

, which implies for

the solution

$\displaystyle X := C_1\cos{kx}+C_2\sin{kx}$

and for

the solution

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

The two first boundary conditions give

, $0 = X(\pi) = 0+C_2\sin{k\pi}$ , and since $C_2 \ne 0$ , we must have $\sin{k\pi} = 0$ , i.e.

$\displaystyle 0 < k := n = 1, 2, 3, \ldots$

Therefore

$\displaystyle X(x) := C_2\sin{nx}, Y'(y) \equiv nD_1\sinh{ny}+nD_2\cosh{ny}.$

The fourth boundary condition now gives that

; thus

and $Y(y) := D_1\cosh{ny}.$ So (1) has infinitely many solutions

$\displaystyle u_n := C_2D_1\sin{nx}\cosh{ny} = A_n\sin{nx}\cosh{ny}$

(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

$\displaystyle u := \sum_{n=1}^\infty A_n\sin{nx}\cosh{ny}$

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

$\displaystyle C = u(x, \pi) = \sum_{n=1}^\infty A_n\sin{nx}\cosh{n\pi} = \sum_{n=1}^\infty(A_n\cosh{n\pi})\sinh{nx}$

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

$\displaystyle A_n\cosh{n\pi} = \frac{2}{\pi}\int_0^\pi C\sin{nx} dx = \frac{2C}{n\pi}(1\!-\!(-1)^n)\quad \forall n\in\mathbb{Z}_+.$

The even

's here give 0 and the odd give

$\displaystyle A_{2m+1} := \frac{4C}{(2m\!+\!1)\pi\cosh(2m\!+\!1)\pi} \quad (m = 0, 1, 2, \ldots)$

Thus we obtain the solution

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

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

Visualization of the solution

**Figure:** Surface plot of the solution , for
$\includegraphics{heat-surface.eps}$

**Figure:** Color-coded plot of the temperature
$\includegraphics{heat-color.eps}$

Remark. The function 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 theory. 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 formula for reduces to the Fourier series

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

for the discontinuous function on $[-\pi, \pi]$ :

$\displaystyle x \mapsto \begin{cases} C\„ & 0 < x < \pi \ -C\„ & -\pi < x < 0 \end{cases}$

That means the Fourier expansion will necessarily be subject to the Gibbs phenomenon. Of course, the series also cannot converge absolutely; in other words, the terms of the series decay too slowly in magnitude, adversely affecting the numerical solution.

Python program to compute and produce the two figures

"example of solving the heat equation" is owned by pahio. [ full author list (2) ]

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stationary example of heat equation

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Cross-references: Fourier series, discontinuous, surface, near, partial sum, half-interval, constant function, Fourier sine series, interval, converges, series, sum, implies, negative, sides, equation, solution, variables, boundary conditions, faces, function
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This is version 20 of example of solving the heat equation, born on 2006-02-22, modified 2008-04-14.
Object id is 7647, canonical name is ExampleOfSolvingTheHeatEquation.
Accessed 6921 times total.

Classification:

AMS MSC:

35Q99 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Miscellaneous)

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