8/14/2019 MIT18_311s09_pset02.pdf
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18.311PrinciplesofAppliedMathematics,S2009. Instructor:
AslanKasimov.Name:
Please make this your title page and write your name in the box
above. Excluding exceptional
circumstances, late homework will not be accepted. Make an extra
effort to explain your solutions with
maximum clarity. Points will be deducted for unclear/incomplete
explanations. AMSYL stands for any
mathematical software you like, such as Matlab, Mathematica,
Maple, etc.
ProblemSet#2(due inclassThursdayFeb. 26).2.1. (15pts) Find
two-term asymptotic expansions of all five roots of the following
innocentlooking quintic:
x5 2x2 + = 0.a) First, try the nave expansion x =x0+x1+2x2+....
to quickly face a difficulty of findingx0, x1, ... To get out of
the trouble, rescale x by figuring out possible dominantbalances
inthe equation and then continue finding the expansions in the new
rescaled variables.
b) For both = 0.1and = 1, find the numerical values of the roots
from your solution inpart a) and compare them against those found
by a root solver in AMSYL. What is theaccuracy of your approximate
solution (i.e. calculate the relative error, assuming the
rootsfound by AMSYL are exact)?
2.2. (15pts) Consider the boundary value problem:x
y = 3y cos , y (0)=0, y(1)=1.2
a) Compute the numerical solution of this problem using AMSYL
and plot it.b) Now solve
xy =ycos , y(0)=0, y(1)=12
by a perturbation method assuming 0to at least two terms, i.e. y
=y0(x) +y1(x) +O (2). Take = 3 in this solution and compare it to
the numerical solution found in parta), i.e. plot the two solutions
together on a single graph. Comment on the accuracy of
theasymptotic solution.2.3. (10pts) Use singular perturbation
methods to obtain a uniform approximate solutionto the following
problem:
y + 2y +y = 0, y(0)=0, y(1)=1.2.4. (10pts) Obtain a uniform
approximation to the solution of the BVP:
y (2x +1)y+ 2y = 0, y(0)=1, y(1)=0.2.5. (10pts) Consider the
initial value problem,
x +x3 +x = 0, x (0)=0, x (0)=1.Assuming the multiple-scale
expansion x =x0(t, ) +x1(t, ) +...., where =t, obtainthe
leading-order solution x =x0(t, ) +O (). Give physical
interpretation of the problemand its solution.
