代写MATH 4700 – Fall 2024 Foundations of Applied Mathematics Practice Midterm代写留学生Matlab语言

Foundations of Applied Mathematics

MATH 4700 – Fall 2024

Practice Midterm

1 Perturbation Theory for Finding Roots to Algebraic Equations (20 points)

If ϵ is a small positive number (0 < ϵ ≪ 1), construct a good approximation for most negative real-valued root x∗ (in the sense that x∗ < x′∗ for any other real-valued root x’∗) of the equation

ϵx3 + 2ϵx2 − 2ϵx + 3 = 0

Your answer should consist of a nonzero main term, the most important nonzero correction term, and an estimate of the error of your approximation. Be sure to fully justify your reasoning. You won’t be penalized if you want to calculate all the roots, so long as you find them correctly, but you won’t get any bonus points either.

2 Perturbation Theory for Finding Roots of Transcendental Equations (20 points plus 10 bonus points)

If ϵ is a small positive number (0 < ϵ ≪ 1), construct a good approximation for all order unity real-valued roots to the equation

e2+ϵ ln(7+3ϵx) = πx + 8

Your answer should consist of a nonzero main term, the most important nonzero correction term, and an estimate of the error of your approximation.

For 10 bonus points give a precise analytical argument for whether or not you expect the equation to have small roots or large roots in addition to the order unity roots you found? Arguments by numerical plots (on the take-home portion) will not by themselves earn much bonus credit.

3 Regular Perturbation Theory for Differential Equations (30 points)

If ϵ is a small positive number, construct a good approximation to the solution of the initial value problem:

Your answer should consist of a nonzero main term, the most important nonzero correction term, and an estimate of the error of your approximation.

4 Motoring With a Shaky Anchor (40 points plus 20 bonus points)

Consider the following mechanical model inspired by a biophysics problem I have been thinking about in my research recently.

Figure 1: Schematic of motor moving to the right along a track, attached by a tether to an anchor making small oscillations across the track.

A “motor” particle moves along a one-dimensional track while tied by a flexible tether to an anchor, with x(t) denoting the position of the motor along the track, relative to the anchor, as a function of time t. The motor velocity is governed in terms of the along-track force F1(x(t), t) pulling back on it by the following equation:

Here v > 0 is the unloaded speed of the motor and fs > 0 is the amount of force necessary to cause the motor to stall (stop moving forward). Meanwhile, the anchor undergoes a small oscillation across the track, with position y(t) = a sin ωt relative to the track, with a and ω some positive parameters. The tether connecting the motor and the anchor is a Hookean spring with spring constant κ and rest length ℓ. The motor starts at the displacement

x(t = 0) = ℓ

along the track at which the tether is relaxed.

An elementary two-dimensional mechanics calculation (which you might wish to pursue in your free time if you like physics, but is beside the point for this exam or class) would show this setup leads to an along-track force on the motor of:

as a function of the along-track displacement x of the motor from the anchor and time t.

a. (15 points) Considering the across-track oscillation of the anchor as a small disturbance, specify and conduct a nondimensionalization for the above equations which is suitable to prepare for a perturbation theory motivated by this assumption.

b. (5 points) Which nondimensional parameter group would you take as a small parameter in the perturbation theory? Explain.

c. (15 points) Develop a perturbation theory for the motion of the motor along the track when the across-track oscillation of the anchor can be treated as small. You should obtain the leading order behavior. explicitly, set up a fully explicit differential equation for a nontrivial correction term due to the small oscillations, and indicate how you would use the solution to this differential equation to obtain an approximation for the motion of the motor along the track. You do not need to solve the differential equation for the correction term; you just need to fully specify it so that it could be, for example, given to your coding assistant to solve numerically once your physics colleague informs your assistant of the actual values of the problem parameters.

d. (5 bonus points) Express the correction term in the perturbation expansion from part c as explicitly as you can analytically.

e. (10 bonus points) (Take-home only) Exhibit the nondimensional form. of your perturbation approximation numerically, choosing values of the nondimensional parameters that are consistent with the problem statement. Make sure the effect of the correction term is clearly visible.

f. (5 bonus points) Over what time scale t would you expect the perturbation theory obtained by combining parts c and d to remain valid? Explain your reasoning.

g. (15 bonus points) The probability the motor has remained on the track up until time t satisfies the equation:

with positive parameters δ, ϕ1, and ϕ2. This evolution depends not only on the along-track force F1(x(t), t) on the motor but also the across-track force F2(x(t), t). The latter can be shown to be expressed as

as a function of the along-track displacement x of the motor from the anchor and time t.

Obtain an approximate expression for the probability that the motor has remained on the track up to a running time t when the across-track oscillation of the anchor can be treated as small. Your answer should involve a main term, a nontrivial correction due to the anchor disturbance, and an estimate for the error of your approximation.

h. (5 bonus points) Over what (dimensional) time scale would you expect your approximation from part g to remain valid? Explain your reasoning.