NO CELL PHONES, HEADPHONES/BUDS, NO CALCULATORS. SAVE FREQUENTLY. SAVE BEFORE PUBLISHING!!!
ONLY MATLAB MAY BE ACTIVE ON YOUR COMPUTER. Acrobat may only be open at the end, when you are ready except for publishing and actively looking at q1.pdf, q2.pdf, or q3.pdf with it.
VARIABLE NAMES MUST BE APPROPRIATE AND CLEARLY SHOW WHAT VARIABLE IT IS OR A 50% CREDIT REDUCTION IS APPLIED. USE THE SAME NAMES AS IN THE QUESTION. All results must have an appropriate understandable name, not ans.
After translation into mathematics, only MATLAB may be used to solve the full problem as posed. Use the appropriate procedures as covered in the lectures.
Open notes: print-outs of lectures and homeworks allowed. You can also use your own homework .m files.
If you cannot get this to work, behind your feeble attempts,
just set manually to the above values to do the rest.
Show both A and its transpose to verify that A is symmetric.
Now let MATLAB find the eigenvalues and eigenvectors of this matrix.
Then put the last two eigenvectors and
in
separate one-dimensional arrays called eVec3 and
eVec4 and put the last eigenvalue
in a scalar
variable lam4. Then use eVec4 and lam4
to verify directly that indeed
.
Also verify that the lengths of both eVec3 and
eVec4 are one, and that eVec3 is orthogonal to
eVec4.
Folium of Descartes
Then define a two-dimensional grid for the plate in which the grid
points have these and
-values. Also define an array
forcing for the grid points that is zero for the grid
points in the interior of the plate. For the grid points on the
boundaries of the plate, array forcing should have the
given temperatures on the boundary: For the left hand and bottom
boundaries the temperature is given to be zero, but on the right
hand boundary it is
and on the top boundary it is
. Put
that in array forcing. Then use array forcing in
function SimplePoisson to create the temperature values
TGrid at all the grid points. (Ignore the warning about
CONDEST.)
(Note: If you cannot get TGrid right, behind your feeble attempts, put the code
xGrid=zeros(m,n); yGrid=zeros(m,n); TGrid=zeros(m,n); for i=1:m; for j=1:n xGrid(i,j)=xVals(j); yGrid(i,j)=yVals(i); TGrid(i,j)=xVals(j)^2*yVals(i); end; endwhich will produce a fake TGrid good enough for doing the rest of this question.)
Plot the obtained temperature distribution as a surface with
smoothly varying shading. Label all three axes, the temperature
axis as Temperature
.
Also plot the isotherms (lines of constant temperature) in the
plane. In particular, plot the lines where the temperature is
0.1, 0.2, ..., and 0.9. Label the axes.
Use double percent lines and/or figure numbers to ensure that both plots end up in the published pdf file.