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AMATH 301 Rahman

Problem 1

In real life we are never given a di↵erential equation to solve. We must ﬁnd the di↵erential 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 @2P P(t = 0,x) = exp ✓1

− ◆ ; 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 di↵erence process we used in lecture, use a second order di↵erence 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 di↵erences, 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 = 1 and save it as A12 (Hint: do you see a pattern as you iterate the problem?)

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 = USVT .

(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.

Problem 4

Consider the boundary value problem

+ x = 5cos(4t),

with x(0) = 1 and x(6) = 0.5.

The true solution to this boundary value problem is

x(t) = C1 sin(t)+ C2 cos(t) − cos(4t),

where

C1 = 2 3 and C2 = .

The solution looks like this:

(1) Following the ﬁnite di↵erence process we used in lecture, use a second order di↵erence scheme to approximate and rewrite this initial value problem as a linear system of equations Ax = b. Use ∆t = 0.1. Note that there will be 61 total t-values, but we are only trying to approximate x at the interior times. The matrix A should be Nint ⇥ Nint, where Nint is the number of interior times, and the vector b should be Nint ⇥ 1. Save a copy of A in a variable named A9. Save a copy of b in a variable named A10.

Solve this linear system using Gaussian elimination (the backslash operator in MATLAB or the solve function in python). Make a 61 ⇥1 column vector containing the x values at all times (including the two boundary values) and save a copy of this vector in a variable named A11.

Find the maximum (in absolute value) error between this approximation and the true solution. Save this error in a variable named A12.

(2) Now solve the problem using the shooting method. Remember, you need to convert this second order ODE into a system of two ﬁrst order ODEs. Apply a bisection scheme to the usual IVP solvers ode45 (MATLAB) or solve_ivp (Python) using ∆t = 0.1. For the second initial conditions, you can use the plot to help you test out values that will get you on di↵erent sides of the right boundary condition. Break out of the loop for bisection when the di↵erence between your approximation at the right boundary point and the exact value at the right boundary point is less than 1e-8; i.e., |xapprox(6) − x(6)| = |xapprox(6) − 0.5| < 10 −8. Save the numerical solution as a 61 ⇥ 1 column vector A13 and the maximum (in absolute value) error max|xapprox − x| as A14. Further save the maximum di↵erence in absolute value between the solution from part 1 and this solution as A15 = max(abs(A13 - A11)).