solutions1.3 - Math 307: Problems for section 1.3 1. Write...

Math 307: Problems for section 1.3
1. Write down the vector approximating f ( x ) at interior points, the vector approximating
xf ( x ) at interior points, and the finite di ff erence matrix equation for the finite di ff erence
approximation with N = 4 for the di ff
1

2. Write down the matrix equation to solve in order to find the finite di ff erence approxima-
tion with N = 4 for the same di ff
3. Use MATLAB/Octave to solve the matrix equations you derived in the last two prob-
lems for the vector F that approximates the solution (i.e., with N = 4 ). Then redo the
calculation with N = 50 and plot the resulting functions.
For define N and Δ x
N=4;
DX=2/N;
Define the matrix L where L F corresponds to f ( x )
L=diag(-2*ones(1,N+1))+diag(ones(1,N),1)+diag(ones(1,N),-1);
L(1,1)=1;
L(1,2)=0;
L(N+1,N)=0;
L(N+1,N+1)=1;
Define the matrix Q where Q F corresponds to xf ( x )
X=linspace(1,3,N+1);
Q=DX^2*diag(X);
Q(1,1)=0;
Q(N+1,N+1)=0;
Define the right side of the equation for the boundary conditions of the first problem.
2

Solve for F
F=(L+Q)\b
The result is
F =
1.00000
0.17778
-0.71111
-1.24444
-1.00000
To solve the second problem we have to change the terms corresponding to the boundary conditions.
L(1,1)=-1;
L(1,2)=1;
b(1)=DX;
F=(L+Q)\b
The result is
F =
1.65385
2.15385
1.84615
0.61538
-1.00000
For the second part of the problem we change N=4 to N=50 above. Then at the end, we can plot the
solution against the correct values of x using
plot(X,F);
Here are the resulting plots:
-1
-0.5
0
0.5
1
1.5
2
1 1.5 2 2.5 3
3

Questions 4–6 deal with the steady heat equation in a one-dimensional rod considered in
the notes:
0 = kT ( x ) - HT ( x ) + S ( x ) ,
where k and H are constants, subject to the boundary conditions
T = T
l
at x = x
l
and T = T
r
at x = x
r
.