Heat equation on a rod

Introduction

Imagine that you have a rod of length L. It could be considered as a 1D heat equation problem and we could draw the rod on the x-axis of the left figure.

The 1D heat equation is in the form

ut = k uxx .

u (x,t)  =   Temperature  (Kelvin)
x  =  Distance from the left side of the rod  (in meters)
t  =  Time  (in seconds)
k  =  Conductivity (in meter*meter/second)

Boundary Conditions:

u (0,t) = Random temperature between 150 and 350 K
u x (L,t) =  0  (or flux free condition.)
where L  =  length of the rod (in meters).

Initial Condition:

u (0,t) = c  where c is a constanst.
(Users could choose c by themselves.)

In this program, we set k =  5  m*m/sec & L = 30 m.

How to use the program

It is easy! Click on the left figure to choose the values of green and gray points on the axes. The green point on x-axis is the point on the rod we use to plot the right figure.) The gray point is the constanst "c" for our initial condition.

What is interesting?

1. You could learn the behaviors of the heat equation which is trying to smooth the graph.

2. Experience with the initial heat "c" which is about, lesser and bigger than 250 K. What does the term "uxx" of the heat equation try to do? How does the random boundary condition effect the solution?  (Hint: On average, the random heat is about 250 K.)

(Notes: The program uses the explicit finite difference method to solve the equation with  dx=L/50  &  dt = 0.49*dx*dx/k .)

Distributed by:

Atichart Kettapun (Noom)
University of California at Santa Cruz
at the MSRI Mathematical Graphics Summer School
at Reed College, Portland, in July 2003