Math 300 -- Directed Readings: RAAA2

Boundary Value Problems for the Heat Equation

1. First, we consider the BVP

and all (this says the endpoints of the thin wire are maintained at 0 degrees,

for example by large blocks of ice)

The Fourier series solution of the BVP is:

where

> assume(n,integer);

> B[n] := (2/5)*int(x*(5-x)*sin(n*Pi*x/5),x=0..5);

As n increases, the exponential terms and the are both tending rapidly to zero.

So we get a good approximation to the solution by including only a few

terms:

> u := (x,t) -> sum(B[n]*exp(-n^2*Pi^2*t/25)*sin(n*Pi*x/5),n = 1..3);

> plot3d(u(x,t),x=0..5,t=0..10);

Physical intepretation: the heat in the rod at time is rapidly drawn out at

the ends and the temperature is tending to zero rapidly at all x as t increases.

2. Another type of BVP. Now we consider boundary conditions , and ,

which correspond to insulating the endpoints of the wire so that our BVP is:

and all (this says the endpoints of the thin wire are insulated so

no heat transfer occurs.)

u(x,0) = x

Solution is

where

> A[n]:=2*int(x*cos(n*Pi*x/5),x = 0 .. 5)/5;

> A[0]:=2*int(x,x=0..5)/5;

> u2 := (x,t) -> A[0]/2 + sum(A[n]*exp(-n^2*Pi^2*t/25)*cos(n*Pi*x/5),n=1..5);

> plot3d(u2(x,t),x=0..5,t=0..10,axes=BOXED,scaling=CONSTRAINED);

Physical interpretation for this one?