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.
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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 , the Math Dept’s new hire . [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 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:
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 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):
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 ofyourchoicefora=b=1andinitialconditionsS(0)=4,R(0)=0.1,R(0)=0,fortheinterval0