Work Hours
Everyday: 北京时间8:00 - 23:59
AMATH301 Rahman Coding Project 10
Due Friday, June 11th at 11:59pm
Problem 1
In real life we are never given a differential equation to solve. We must find the differential equation based
on what we know about the process and physical laws. Let’s look at the tank problem that we did in Week
8, but this time with a slight twist.
Two tanks with capacities of 10 liters initially contain 2 grams of salt and 1 liter of water. Water containing
1 g/L of salt enters the first tank at a rate of 2 L/hour, and the well-mixed solution flows out of the first tank
into the second tank at a rate equivalent to the volume of the first (i.e., V1 L/Hr). None of the brine flows
out of the second tank.
To model the process follow the steps in the lecture for each tank separately. Notice that the rate for the
second tank will depend on the first.
(1) Model the problem for the volume in liters as an IVP(s) and by using ode45 (for MATLAB) or
solve_ivp (for Python) solve the IVP (ODE + IC) for the volume of water in the tanks. Solve from
time t = 0 to t = 6 with ∆t = 0.01. Save the volume of the first tank at each time as a 601 ×1 column
vector named A1 and the volume of the second tank at each time as a 601 × 1 column vector named
A2.
(2) At what time to the nearest integer does the water start overflowing (hint: use the round function)?
Save this integer value as A3. And which tank does it overflow from (1 or 2)? Save this integer value
as A4.
(3) Model the problem for the amount of salt in grams as an IVP(s) and by using ode45 (for MATLAB)
or solve_ivp (for Python) solve the IVP (ODE + IC) for the amount of salt in the tanks. Solve from
time t = 0 to t = 6 with ∆t = 0.01. Save the amount of salt of the first tank at each time as a 601 × 1
column vector named A5 and the amount of salt of the second tank at each time as a 601 × 1 column
vector named A6.
(4) What is the salt concentration when tank that overflowed in part (b) is completely full of brine? Save
this salt concentration (remember it’s amount/volume) as A7.
Problem 2
Suppose the initial probability of finding a quantum particle on the line from x = −1 to x = 1 is a compact
Gaussian. The particle has equal probability of going to the left or to the right, and has the ability to leave
the region. What is the probability that the particle is located at x = 0.5 at t = 1?
This can be solved via the following PDE:
∂P
∂t =
∂
2P
∂x2
; P(t = 0, x) = exp
1 −
1
1 − x
2
; P(t, x = −1) = P(t, x = 1) = 0. (1)
We will use the Crank-Nicolson scheme to numerically solve the problem.
(1) Following the finite difference process we used in lecture, use a second order difference scheme to
approximate Pxx and rewrite this as a linear system of equations Ax = b. Here the matrix A will
be the usual tridiagonal matrix from finite differences, and the vector b will change at each timestep.
Use ∆t = 0.01 and ∆x = 0.01. Save a copy of A in a variable named A8. Save a copy of b as a 199×1
column vector after the first iteration in a variable named A9.
Solve this linear system at each time t = [0:0.01:1] using Gaussian elimination (the backslash
operator in MATLAB or the solve function in python). Save a copy of b as a 199 × 1 column vector
after the last iteration in a variable named A10.
Save the probability of finding the particle at x = 0.5 at t = 1 as A11. Find the exact probability of
finding the particle at x = 0.5 at t = ∞ and save it as A12 (Hint: do you see a pattern as you iterate
the problem?)
1
Problem 3
Download the files CP5_M1.mat, CP5_M2.mat, CP5_M3.mat from our Canvas page. Then load the files onto
MATLAB with the following commands:
load CP5_M1.mat
load CP5_M2.mat
load CP5_M3.mat
or Python with the following commands:
After running these commands the variables M1, M2, and M3 will be defined. It is your task to figure out
which one represents each of U, S and V. [Hint: You should be able to see a clear picture]. After you find
which matrices are U, S, and V, let data = USV T
.
(1) Calculate how many megabytes it would take to store the matrix data. That is, calculate how many
entries there are in this matrix and multiply by 8 / 1e6. Save your answer in a variable named A13.
(2) Find the number of singular values that capture more than 99% of the information in this image. That
is, find the smallest value of k such that
Ek = (m ∗ k + k + n ∗ k)/(m ∗ n) > 0.99,
where m is the number of rows and n is the number of columns of data. Save the number you found
for k in a variable named A14.
(3) Calculate how many megabytes it would take to store this new matrix the same way you did in part
(1). Save your answer in a variable named A15.
(4) Use the Week 10 coding lecture to guide you in displaying the image that this SVD encodes using
the uint8 and imshow commands on MATLAB. Save the integer corresponding to the word that best
describes this image as A16.
1) porter, 2) pig, 3) profit, 4) quiet, 5) quicksand, 6) cast, 7) quince, 8) fang, 9) aftermath, 10)
experience, 11) ticket, 12) linen, 13) creator, 14) look, 15) trouble, 16) teeth, 17) dog, 18) arm, 19)
basin, 20) toad, 21) dinner, 22) position, 23) snail, 24) snake, 25) alarm, 26) ocean, 27) act, 28)
argument, 29) cable, 30) cattle, 31) ink, 32) reward, 33) pump, 34) university, 35) church, 36) kittens,
37) government, 38) umbrella, 39) doll, 40) secretary, 41) spring, 42) bird
It may also be instructive to reconstruct the image using 1,2,3,… of the singular values and see how
many are really needed to fully capture the image. Since this part is subjective, it will not be part of
the submission.
https://amath.washington.edu/courses/2022/spring/amath/301/a
