# Solution of the Laplace Equation for Temperature Distribution in a Square

Solution of the Laplace Equation for Temperature Distribution in a Square

Consider the Laplace equation u+u=0 in a square region, where . We wish to solve for the temperature distribution , subject to the following Dirichlet boundary conditions:

2

∂

∂

2

x

2

∂

∂

2

y

0⩽x,y⩽1

u(x,y)

BC1: ,

u(x=0,y)=1

BC2: ,

u(x=1,y)=0

BC3: ,

u(x,y=0)=0

BC4: .

u(x,y=1)=0

This Demonstration solves the problem using a spectral method [1] for a user-set value of the number of the Chebyshev–Gauss–Lobatto collocation points. The colored dots reflect the intensity of the scalar field at the various nodes in the square domain. You can clearly recognize the form of a contour plot of in the third snapshot.

u

[0,1]×[0,1]

u