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###### 日期：2020-12-10 10:15

Fall 2020 ME6200 Math Methods:

Assignment - 8

Due: Mon. Dec. 7 (Start of lecture)

Problem 1) 17.6.1 (The “Matt Damon” problem… In the movie “Good Will Hunting”, the

solution to problem 17.6.1 is on the blackboard at the 0:28 mark in this clip: http://

www.math.harvard.edu/~knill/mathmovies/swf/goodwillhunting.html. Good luck

deciphering it.) Note that the error defined in problem 17.6.1 is like an average error

between the series and the function one is trying to represent with the series… if one is

interested in the “worst case” error rather than average error, there are different

relations one could derive.

Problem 2) 17.6.2.a, b, g.

Problem 3) Repeat example 17.7.3, but change the BC on the left hand side at x=0 to

y-y’=0 instead of y-2y’=0. Find the first 6 eigenfunctions/eigenvalues numerically using

the FindRoot command in Alpha/Mathematica or the fzero command in MATLAB. You

will need to provide reasonable initial guesses for the numerical solution for the

eigenvalue, and you should make this choice by generating a plot analogous to Fig

17.7.3. (Fourier didn’t have this option.). Alternatively, you can keep zooming in to the

zero-crossings by hand with the “plot” command to locate the roots.

Problem 4). Use the first six eigenfunctions from problem 3 to represent the function,

f(x)=x on the interval x<0<1. Note that the function, f(x)=x, doesn’t satisfy the same

BCs as the eigenfunctions of the SLP, but we can still use the SL-series to represent

it… that’s one of the main points of Fourier/Sturm-Liouville methods, and we will see

this again and again in chapters 18-20.

Problem 5: Greenberg 17.8.3.

Note for 17.8.3 that H(x) is the Heaviside unit-step function: H(x)=0 for x<0, H(x)=1 for

x>0. Use projection techniques and recall that the weight function for the given SL

problem is w(x)=1 and the eigenfunctions are the Legendre polynomials, Pn(x),

discussed in chapter 4. Feel free to use Wolfram Alpha, Mathematica, or MATLAB to

compute the projection integrals. The command to access the Legendre polynomials is

LegendreP[n,x] in Mathematica/Alpha and legendreP(n,x) in MATLAB.

Problem 6: Greenberg 18.3.9. This is a “meat and potatoes” Dirichlet BC problem.

(Expect something like this on Exam3.)

Problem 7: Modify Greenberg 18.3.9. to use a different BC so that u=0 on the left at

x=0 and u’=0 on the right at x=2 (the left is kept at fixed temperature, the right is

insulated). Use the same initial condition as in Greenberg’s 18.3.9.a.

Problem 8: 18.3.13.

Problem 9: Greenberg 18.3.15, 16a, 16b (Inhomogeneous equation with Dirichlet BCs).

Physically, if we are thinking of diffusion of heat with u equal to the temperature, then

the -F term is like having a refrigeration along the length of the bar removing a

prescribed amount of heat per unit time (power) per unit length.

Problem 10: Repeat example 18.3.4, but let the BC on the left side (at x=0) be: u-u’=5

and the BC on the right side (at x=1) to be u=40. (This utilizes the SL problem you were

asked to solve numerically on the previous HW assignment). Take the specific case of

the initial temperature profile to be spatially constant u(x,t=0)=20.

a) Numerically find the first 3 eigenvalues for the Sturm-Liouville series. (You may use

the simply restate results from problem 3/4.)

b) Approximate the solution to the initial value problem using only the first three

eigenfunctions of the Sturm-Liouville series.

c) Plot your approximate, 3-term truncated, solution at the following times: t=0.1, t=0.5,

t=1, t=2. (Note, for simplicity we have set α2=1 and L=1. Use the Plot command in

alpha.wolfram.com, Mathematica, or the equivalent plot command in MATLAB)