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Web Programming

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?)

Sample Solution

Another relationship that played a role in prompting Desdemona’s death was the one with Michael Cassio. Through malicious manipulations carried out by Iago, Cassio was presented to Othello as the one his wife had a secret affair with. However, whilst most of it was indeed the product of Iago’s actions, some aspects that made Othello believe in the possibility of such an affair were to be blamed on Desdemona herself. Unlike with Brabantio, Desdemona shows extreme simplicity in her relationship with Cassio, who actively abuses their friendship. For some reason, Desdemona does not notice Cassio’s intentions to exploit her position of “the Captain of the Captain” and get a promotion in the army. Thus, she blindly accepts Cassio’s request to persuade Othello, which, in fact, would be an interference with state affairs. Conspicuous is also Desdemona’s eagerness to fulfill her mission. She is very insistent in her intent to get Cassio promoted. This is reflected in the scene where she holds a conversation with Othello regarding Cassio’s audience, and persistently repeats her quest seven times: “But shall’t be shortly?”, “Shall’t be tonight, at supper?”, “Tomorrow dinner then?”, “Why, then, tomorrow night, or Tuesday morn, / On Tuesday noon, or night; on Wednesday morn / I prithee name the time, but let it not / Exceed three days.” (III, iii, 56-63) To an extent such demand could be perceived by Othello as an indicator of extreme loyalty to Cassio, especially as opposed to the extreme lack of the latter to her own father, and subsequently raise questions about the nature of it. It is also important to consider the relationship Desdemona had with Othello. Although it is undeniable that love between the two existed, it can be argued that they were not particularly aware of each other’s characters due to the rather short period of wooing before the marriage took place. At least, Desdemona was not, as despite the continuous hints regarding what Othello had prepared for her, she decides to ignore it. Early in the play Desdemona fails to accept the fact that unlike her, Othello is very much interested in how things appear to the general public, and not only how they really are. The concept of infallibility of reality is so strong in Desdemona’s mind, she continues to ignor
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