HW8

More matrices from "real" problems.  We continue to build 
a collection of real matrices for real problems. These
are the examples discussed in class and are mostly
also discussed in the book p190-207.
 

1) Write a function that generates and 
solves the appropriate 
linear system to compute the FDM approximation 
of the PDE

u_xx + u_yy + u_zz = x + y + z^2

on the domain 

	0<x<1
	0<y<1
	0<z<1

with the zero Dirichlet boundary conditions 
on the boundary of the domain.  As a test you
should use a 3D grid with 11 points in each 
direction but your code should be capable of 
solving the same problem with a different 
number of points.  Use the built-in solver
to solve the linear system.

2) Write a function that generates and solves the
appropriate linear system to compute the FEM
approximation to the PDE

-u_xx = sin(x^2)

on the domain

	0<x<1

with homogeneous Dirichlet boundary conditions

	u(0)=u(1)=0

using the set of shape functions
sin(n Pi x) 1<n<10.

3) Write a function that generates and solves the
appropriate linear system to compute the FEM
approximation to the PDE

-Lap u = Theta sin(r^2)

on the radial domain

        0<r<1


with homogeneous Dirichlet boundary conditions

        u(1,theta)=0 and u(0,theta)-0

using the set of shape functions with (0 <= m <= 4)
	(1-r) r^m cos(n theta) 0 <= n < 4
and
	r^m sin(n theta) 0 < n < 4.