# Gibbs Phenomenon in Laplace's Equation for Heat Transfer

Gibbs Phenomenon in Laplace's Equation for Heat Transfer

This Demonstration plots the solution to Laplace's equation for a square plate, u(x,y)+u(x,y)=0.

2

∂

∂

2

x

2

∂

∂

2

y

The solution is given by

u(x,y)=sinysinh(x-L)

∞

∑

n=1

A

n

nπ

H

nπ

H

where and are the length and height of the plate (here ) and

L

H

L=H=1

A

n

2

Hsinh

nπ[x-L]

H

H

∫

0

nπy

H

where .

g(y)=T

You can vary the temperature along the left edge. As you increase the number of terms , observe the Gibbs phenomena at the corners and along the edge where the temperature is higher.

T

n