Steady state temperature and Laplace's equation

Temperature is defined by a given function around the edge of a two-dimensional plate. What will be the steady-state temperature within the plate?

This problem is discussed in Section 9.7 of Edwards & Penney (1996). The following two images show the steady-state temperature distributions for Examples 1 and 2. The second example has been rotated so it can be compared with the first. The first example is a square plate with the temperature held at 100^oC along the lower edge and at 0^oC along the remaining edges. The second is the bottom of a semi-infinite strip with the temperature held at 100^oC along the lower edge and at 0^oC along the left and right edges.

Temperature scale: 0^oC 100^oC

Example 1 has solution

u(x,y) = [4 T₀ / pi] sum_{n odd} {sin (n pi x/a) sinh (n pi [b - y]/a) / [n sinh (n pi b/a)]}

where a = 10 is the width of the plate and b = 10 is the height of the plate. Example 2 has solution

u(x,y) = [4 T₀ / pi] sum_{n odd} {1/n exp (-n pi x/b) sin (n pi y/b)}

where b = 10 is the width of the semi-infinite strip. The accuracy of the solution depends on the number of terms used in series. The following images show approximations with 1, 2, 3, 4 and 5 terms. The larger images were calculated with 6 terms.

Square plate

Semi-infinite plate

Numerical approximations to more general temperature distributions can be found using the Jacobi iteration. First, the plate is divided into an array of cells and the temperature of each boundary cell set to the appropriate value. The interior cells are then given an initial value which is improved using the iteration

u_n+1(i,j) = [u_n(i-1,j) + u_n(i+1,j) + u_n(i,j-1) +u_n(i,j+1)]/4

where u_n(i,j) is the temperature of cell (i,j) at iteration n.

The first example has u = 0 along the upper edge, left edge and right edge, and u = 100 along the lower edge. The two lower corners have u = 50. The interior points are initially set to u = 0. The animation shows 120 Jacobi iterations.

The next example has u = 0 along each edge, and an `L' shape in the middle with u = 100.

The final example has sinusoidal temperature distributions along the upper and lower edges, and linear temperature distributions along the left and right edges.