ME6200编程辅导、讲解MATLAB程序语言、MATLAB编程辅导讲解SPSS|讲解数据库SQL

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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)

ME6200编程辅导、讲解MATLAB程序语言、MATLAB编程辅导 讲解SPSS|讲解数据库SQL

ME6200编程辅导、讲解MATLAB程序语言、MATLAB编程辅导讲解SPSS|讲解数据库SQL