matlab代写-4B
时间:2022-03-29
Lab4B (Version 2)
Numerical solution of Differential Equations & Application Area

In Lab4B will explore methods for solving ODE numerically and apply these methods. Sources for this Lab
include sections 6.1, 6.2 and parts of 6.3 and 6.4. You are also encouraged to find your own sources.
Some additional resources will be made available. Labs will go ahead as usual this week and the
following week.

You will write an integrated essay Report, in the same format as the previous labs. The report +
appendices etc should be submitted to Gradescope. It should cover the following aspects.

Title, team members, Abstract, Introduction (max 1 page)

Section 1 Introduction

Section 2 Numerical Solution of ODE (max 2 pgs)

This section is about the numerical solution of ODE. It should include a brief discussion of the main
methods you encountered, accuracy and why the methods are important, and be illustrated by
examples. As usual you should integrate a storyline that connects this section with the rest of the
report, and your chosen focus in Section 3.

Section 3 Application of Numerical Solution of ODE to [You choose one of Q3, Q4, Q5, Q6, Q7]

For completeness the Problem Areas Q3 to Q5 are outlined here. As updates are made to these
Problem Areas, they will be noted in revised versions of this document.


3-Q3: Improvements of Newton’s Method with application to Fractals

Part of this question is developing more fractal graphics, extending your earlier lab work. This involves producing
fractal graphics more complicated polynomials and improving Newton’s method to make better graphics.

One aspect that will be improved for Newton’s method is guaranteeing it numerically approximates all n roots of
an n-th degree polynomial p(z) = 0. This method starts from the n known roots of an easily exactly solvable related
polynomial, then numerical solution of an ODE is executed to numerically follow paths from the known roots to all
n roots of the difficult polynomial. In particular, start with the easy polynomial q(z) = z^n – 1 = 0 and give the roots
in terms of complex exponentials (see an earlier lab) for n = 2 and n=3. Suppose that p(z) = 0 is the difficult
polynomial and you want to approximate all the roots of p(z) = 0. Then let H(t) = (1-t)*(z(t)^n – 1) + t*p(z(t)) = 0.
Show that H(0) = z^n – 1, and H(1) = p(z). Show that the derivative of H(t) gives
H’(t) = ( (1-t)*n*z(t)^(n-1) + t*p’(z(t)))*z’(t) + p(z(t))- (z(t)^n – 1) = 0 (H-Eqn)
Then solve for z’(t) to show that you get
z’(t) = ((z(t)^n – 1) - p(z(t)))/((1-t)*n*z(t)^(n-1) + t*p’(z(t))) = F(t,z(t)) (H-DEqn)
Then to approximate all the roots of p(z) = 0 use a numerical ODE solver for t = 0 .. 1 with initial conditions z(0)
equal to each of the exactly known roots of q(z) = z^n – 1 = 0. The output at t=1 are the desired roots. Give
examples for n = 2, 3 and higher if you can.
Though quite simple this intersects with research of Siyuan Deng, Greg Reid the Math Dept’s new hire
Taylor Brysiewich. [I suspect that he changed his name to Taylor since he is researching Taylor-Newton methods!].

3-Q4: Hodgkin-Huxley model for Neurons (How the Giant Squid Axon changed Neuroscience)

Our text describes Hodgkin-Huxley’s model of Neuron dynamics for which they were awarded a Nobel prize. It was
the birth of the new area of Computational Neuroscience. For a short video intro see:
https://www.youtube.com/watch?v=dxbffhJWd7M
At UWO we have the world leading Brain and Mind Institute, and math neuroscientist Lyle Muller in the math
dept: https://www.uwo.ca/bmi/ https://mullerlab.ca
Skills such as numerical ODE solving in Brain Science are in demand! For example, Lyle is looking for talented
undergrad students). The text also has a Runge-Kutta based implementation for solving the giant squid axon model
which is described in 6.4.2. For the code for the model see:
https://media.pearsoncmg.com/aw/aw_sauer_num_analysis_3/code/hh.m

3-Q5: Collapse of the Tacoma Narrows Bridge
A fascinating application of DE solving is to the famous collapse of this bridge. See:
https://www.youtube.com/watch?v=j-zczJXSxnw
The Text has a nice DE model for illustrating numerical solution applied to this bridge (see 6.4, Reality Check 6: The
Tacoma Narrows Bridge (pgs 337-340). For code for the model see:
https://media.pearsoncmg.com/aw/aw_sauer_num_analysis_3/code/tacoma.m
3-Q6: Chaos, Strange Attractors and Climate Change
See the Text description of the Lorenz Equations a system of ODE to illustrate some aspects of climate modeling
introduced by Lorenz [Section 6.4.3, and Computer Problems 6.4/11, 12, 13.] Use the Text’s Matlab for the Lorenz
system to reproduce some of their calculations in Figure 6.17 in 6.4.3. Then do problems 11 and 13 in 6.4, and
perhaps try 12. Discuss the meaning of chaos for this example. As part of this report you should find some nice
references for chaos and the Lorenz attractors.
Numerical solution of such ODE shows chaotic behavior. Watch the video below: How can climate be predictable
if weather is chaotic? See: https://www.youtube.com/watch?v=i5fwYtU7Rhg
3-Q7: SIR model for virus transmission
Let S(t) (susceptibles), I(t) (infectives) and R(t) (recovered) be functions of time (t). The SIR model is a system of
differential equations representing these quantities.
Watch the following short video easy intro to the SIR model:
https://www.youtube.com/watch?v=XWXqXzAYe4E
The following video should be viewed after the previous video. It is a very nice intro to the derivation of the
model DEs (especially the first half of the video):
https://www.youtube.com/watch?v=NKMHhm2Zbkw
The SIR model has the form
S’ = - a*S*I, I’ = a*S*I – b*I, R’ = b*I
where the positive quantities a (the reproduction factor or the average number of contacts per person per unit time)
and b are determined from data. In particular obtain the numerical solution of this system with a numerical method
of your choice for a = b = 1 and initial conditions S(0) = 4, R(0) = 0.1, R(0) = 0, for the interval 0 < t < 6. Plot a
graph for the approximate S(t), I(t), R(t) over this interval. Part of your description should involve descriptive bio
compartment-modeling in terms of interactions between the populations S, I and R. The Math Dept has expert Bio-
Modelers (Lindi Wahl and Geoff Wild) and Ordinary Differential Equation Systems experts (Pei Yu and Xingfu
Zou) who welcome talented undergraduate students. In reality the quantities a and b change with time. For example
new variants such as Omycron have higher infectiousness, and also changes in public policies (such as mandates on
social distancing and masks) affect infection rates and hence a and b. In the next stage we ask you to now declare a
and b to be functions of t. Introduce your own piece-wise defined functions for a(t) and b(t), and execute a
simulation over the same time interval. Interpret your results.
Next page

Lab4B Choice of Application (3-Q3 – 3-Q7)

Your Team will by 11:59 PM, Tuesday March 22, select & post on Gradescope Assignment:
ChoiceOfApplicationLab4B

Lab 4B Video (5 min max) There will be a separate link for submission of the Video
• All teams must at least introduce themselves: give an audio introduction with your
name and face visible if possible. Say hello. Maybe share some thing about yourself …
• Selected Highlights (do not try to cover everything you did)
• Complete coverage is impossible (and not advised)
• Roughly equal time for team members
• Enjoy yourself �
• Submit your videos at the Gradescope link SubmitYourLab4BVideo here (.mp4 format)
Lab 4B Oral Q&A about Lab4B (5 min max)
• On Zoom in your Lab time
• Roughly equal time for team members
• In your designated Lab4B time - reasonably low key
• General questions about your lab (an encouragement)


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