hmk#8_s (1).pdf - MATH 241 - Partial Differential Equations...

MATH 241 - Partial Differential Equations ( Homework # 8 )
Fall Semester, 2020 M. Carchidi
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Problem # 1 ( 25 Points ) - Why Not A Regular Sturm - Liouville Problem
a.) ( 5 points ) While solving a regular Sturm-Liouville Problem a student found eigenvalues

n
satisfying the equation 100 
n
sin  1/ 
n
  1. What is wrong with this result?
b.) ( 5 points ) While solving a regular Sturm-Liouville Problem another student found
eigenvalues given by 
n
 n 2  and corresponding eigenfunctions given by
u
n
 x   sin  
n
x
3

for n  1,2,3, … , and 0  x  1. What is wrong with this result?
c.) ( 5 points ) While solving a regular Sturm-Liouville Problem another student found
eigenvalues given by 
n
  n and corresponding eigenfunctions given by
u
n
 x   sin  
n
x 
for n  1,2,3, … , and 0  x  1. What is wrong with this result?
d.) ( 5 points ) While solving a regular Sturm-Liouville Problem another student found
eigenvalues 
n
to satisfy the equation 
n
sin  1/ 
n
  cos  1/ 
n
 . What is wrong with this
result?
e.) ( 5 points ) While solving a Regular Sturm-Liouville Problem on the interval 0  x  1, a
student claims to have determined 
n
 n for the eigenvalues, with corresponding
eigenfunctions

n
 x   sin
2
 n  x 
for n  1,2,3, .... Explain why the student must be mistaken.
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Problem # 2 ( 20 Points ) - Orthogonal Series
a.) ( 10 Points ) Show that the functions 
n
 x   sin  n  x 2  , for n  1,2,3,..., form are
orthogonal set with respect to the “dot” product
f  g 

0
1
f  x  g  x  xdx .
b.) ( 10 Points ) Use this property to then determine an expression for c
n
if a function h  x  is
expanded as
h  x  
∑
n  1

c
n

n
 x 
for 0  x  1 and compute the c
n
’s for the special case when h  x   x
2
. Then make plots
of h  x  against
h
m
 x  
∑
n
m
c
n

n
 x 
for 0  x  1 and for m  1,2,3,4,5,10 and 20 and comment on your results.
 1
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Problem # 3 ( 15 points ) - Does This Contradict Sturm - Liouville Theory
Consider the boundary-value problem: u ′′  x    u  x   0, for 0  x  1, with u  0   0
and u
′
 1  −  u  1   0.
a.) ( 10 points ) Show that the eigenvalues to this boundary-value problem are the positive
solutions to the equation cot  
n
  
n
, and the corresponding eigenfunctions are
u
n
 x   sin  x 
n
 , for n  1,2,3, … .
b.) ( 5 points ) By direct calculation, show that

0
1
u
n
 x  u
m
 x  dx ≠ 0
for n ≠ m . Is this a contradiction to Sturm-Liouville Theory? Explain.
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Problem # 4 ( 20 points ) - A Regular Sturm - Liouville Problem
a.) ( 10 points ) Determine the eigenvalues ( 
n
) and eigenfunctions ( 
n
 x  ) for the differential
equation
 ′′  x  − 2 x  ′  x      x 2    x   0
for 0 ≤ x ≤ 1, along with the boundary conditions,  ′  0     1   0. Hint : See Problem
#4 of Homework #4.