418/518 Test #3
Study Guide
Page 1 of 3
Problem #1:
Use the Rayleigh Quotient
 pu
RQ[u ] 
L
L
  du  2

du
   p    qu 2 dx
dx 0 0   dx 

L
  u dx
2
0
to obtain an upper bound for the smallest eigenvalue of following boundary value
problem:
( p ') ' [  q]  0,
a (0)   '(0)  0,
 ( L)   '( L)  0
where  ,  ,  ,  are given constants, and p( x), q( x),  ( x) are given functions.
(Assume that p( x),  ( x) 0 and that p( x), q( x) and  ( x) are continuously differentiable.)
Hint: Use a trial function of the form u  A  Bx  Cx 2 that satisfies the given boundary
conditions.
Read section 5.6, pp. 189-192, especially the example on page 191-192.
Do prob. # 5.6.1 (a), (b), (c) on page 194.
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Problem #2:
Use the Rayleigh Quotient
L
L
  du  2

du
 pu
   p    qu 2 dx
dx 0 0   dx 

RQ[u ] 
L
2
  u dx
0
to obtain upper and lower bounds for the smallest eigenvalue of the following boundary
value problem:
( p ') ' [  q]  0, a (0)   '(0)  0,  ( L)   '( L)  0
where  ,  ,  ,  are given constants, and p( x), q( x),  ( x) are given functions.
(Assume that p( x),  ( x) 0 and that these functions are continuously differentiable.)
Read section 5.7, pp. 196-197. Do problem 5.7.1 and 5.7.2 on page 198.
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Problem #3:
Solve the initial boundary value problem
u  
u 
(x)   K 0 (x)  (t  0,0  x  L),
t x 
x 
u(x,0)  f (x) (0  x  L),
l
 ux (0)   u(0)  0,  ux (L)   u( L)  0
where each boundary condition is of the first or second kind, i.e.
  0,   1,   0,   1 or   1,   0,   0,   1 or   1,   0,   1,   0 .
(a)-[15 pts. ] Find a sequence of product solutions of the form un ( x, t )  hn (t )n ( x) that
satisfy the given boundary conditions.

(b)-[10 pts. ] Determine the coefficients in the series u ( x, t )   a n un ( x, t ) so that
n 1


n 1
n 1
u ( x, 0)  f ( x)   a n un ( x, 0)   a n n ( x) .
Read section 5.2, pp. 158-161. Read sections 5.3 & 5.4, pp. 161-172. Do prob. #5.3.9
on page 169. Do probs. # 5.4.2-5.4.3 on page 173. Read Supplement II, especially the case
where   1 .
1
Hint: In the test problem K 0 (x)  x and (x) 
x
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418/518 Test #3
Study Guide
Page 2 of 3
Problem #4:
Solve the following initial-boundary value problem:
u
 2u
 k 2 (t  0, 0  x  L, k 0),
t
x
u ( x, 0)  f ( x), ut ( x, 0)  g ( x) (0  x  L),
u x (0)  h1u (0)  0, u x ( L)  h2u ( L)  0 (h1 , h2  0, h1  h2 0)
(a)-[15 pts.] Find a sequence of product solutions of the form un ( x, t )  hn (t )n ( x) where
n ( x) (n  1, 2, 3, ...) are the eigenfunctions of the regular Sturm-Liouville problem
 ''   0 (0 x 1),  '(0)  h1 (0)  0,  '(1)  h2 (1)  0 . In answering this question
derive an equation satisfied by the eigenvalues of this problem. Use this equation to
derive an upper and lower bound for the smallest (positive) eigenvalue.

(b)-[10 pts.] Determine the coefficients in the series u ( x, t )   a n un ( x, t ) so that
n 1


n 1
n 1
u ( x, 0)  f ( x)   a n un ( x, 0)   a n n ( x) .
Read section 5.8, pp. 198-201, and 204-209. Do probs. #5.8.1, 5.8.3, 5.8.5, 5.8.6 on pp. 210-211.
Also do problem 5.8.8 on page 210.
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Problem #5:
Use the method of separation of variables to solve the initial-boundary value
problem
 2
 2u
 2u 
2  u

c

 x 2 y 2  (t  0,0  x  L,0  y  H ,c 0),
t 2


u(x, y,0)  f (x, y), ut (x, y,0)  g(x, y) (0  x  L,0  y  H ),
1ux (0, y,t)  1u(0, y,t)  0,  1ux (L, y,t)  1u(L, y,t)  0 (0  y  H ),
 2u y (x,0,t)  2u(x,0,t)  0,  2u y (x, H ,t)   2u(x, H ,t)  0 (0  x  L),
where each of the given boundary conditions are of the first or second kind, i.e.
1  0, 1  1,  1  0, 1  1, or
 2  0,  2  1,  2  0,  2  1, or
1  1, 1  0,  1  0, 1  1, or , and  2  1,  2  0,  2  0,  2  1, or .
1  1, 1  0,  1  1, 1  0,
 2  1,  2  0,  2  1,  2  0.
(a)-[15 pts.] Find a sequence of product solutions umn ( x, y, t )  hmn (t )mn ( x, y) of the
given partial differential equation that satisfy the given boundary conditions.
Note that mn (x,y)  fn (x)gm (x) where the functions fn (x) are the eigenfunctions of the regular Sturm-Liouville problem
f ''   f  0, 1 f '(0)  1 f (0)  0,  1 fx (L)  1 f (L)  0 (0  x  L),
and the functions g m (y) are the eigenfunctions of the regular Sturm-Liouville problem
g" g  0,  2 g'(y)  2 g(y)  0,  2 g'(y)   2 g(y)  0 (0  y  H).
We note that mn (x,y)  mn (x,y) where mn   n   m . Furthermore, hmn (t)  Amn cosc mn t  Bmn sinc mn t, which implies th
hmn (0)  Amn ,and h'mn (0)  c mn Bmn .
(b)-[10 pts.]
'
Determine the constants in hmn (0) and hmn
(0) so that the series hmn (t)mn (x, y)
m,n
converges to u ( x, y, t ) , the solution of the given partial differential equation that satisfies the initial
'
conditions, i.e. u ( x, y, 0)   ( x, y )   hmn (0)mn ( x, y ) , and ut ( x, y,0)   ( x, y)   hmn
(0)mn ( x, y) .
m,n
m,n
Read section 7.3, pp. 280-286. Do prob. # 7.3.1 (a)-(d), and 7.3.4 pp. 286-287.
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418/518 Test #3
Study Guide
Page 3 of 3
Problem #6:
Solve the following boundary value problem for Laplace’s Equation:
u  u xx  u yy  u zz  0 (0  x  L, 0  y  H , 0  z  W ),
1u x (0, y, z )  1u (0, y, z )  0,  1u x ( L, y, z )   1u ( L, y, z )  0 (0  y  H , 0  z  W ),
 2u y ( x, 0, z )   2u ( x, 0, z )  0,  2u y ( x, H , z )   2u ( x, H , z )  0 (0  x  L, 0  z  W ),
 3u z ( x, y, 0)   3u ( x, y, 0)  0,  3u z ( x, y,W )   3u ( x, y,W )  f ( x, y ) (0  x  L, 0  y  H ),
where each of the given boundary conditions are of the first or second kind, i.e. where
1  0, 1  1,  1  0, 1  1, or
1  1, 1  0,  1  0, 1  1, or
1  1, 1  0,  1  1, 1  0,
and
 2  0,  2  1,  2  0,  2  1, or
 2  1,  2  0,  2  0,  2  1, or
 2  1,  2  0,  2  1,  2  0,
and where
 3  0,  3  1,  3  0,  3  1, or
 3  1,  3  0,  3  0,  3  1, or
 3  1,  3  0,  3  1,  3  0.
(a)-[15 pts.] Find a sequence of product solutions of the form umn (x, y, z)  hmn (z) mn (x, y) that satisfy
the given homogeneous boundary conditions
1u x (0, y, z )  1u (0, y, z )  0,  1u x ( L, y, z )  1u ( L, y, z )  0 (0  y  H , 0  z  W ),
 2u y ( x, 0, z )   2u ( x, 0, z )  0,  2u y ( x, H , z )   2u ( x, H , z )  0 (0  x  L, 0  z  W ),
 3u z ( x, y, 0)   3u ( x, y, 0)  0, (0  x  L, 0  y  H ).
We note that the functions mn (x, y) are as defined in Problem #5. Furthermore,
hmn (z)  Amn cosh mn z  Bmn sinh mn z,
where the constants Amn and Bmn are multiples of each other chosen so that  3
'
hmn
(0)
  3hmn (0)  0.
dz
Note that hmn (0)  Amn , and h'mn (0)  mn Bmn .
(b)-[10 pts.] Finally, determine the constants in hmn ( z ) so that
 3uz (x, y,W )   3u(x, y,W )  f (x, y)   ( 3
m,n
'
hmn
(W )
dz
  3 hmn (W ))mn (x, y) (0  x  L,0  y  H ).
Read section 7.3, pp. 280 -286. Do prob. # 7.3.6 and 7.3.7 (a)-(b), and 7.3.4 (a)-(d) on pp. 287-288.
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