418/518
STUDY GUIDE-FINAL EXAM
12/11/08
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Problem #1:
Consider the following initial-boundary value problem for the 1-dimensional heat
equation:
(1)  ( x)ut  ( p( x)ux ) x  q( x) (1  x  e, t 0),
(2)-(a)  u (1, t )   ux (1, t )  A, (2)-(b)  u (e, t )   u x (e, t )  B (t
0),
(3) u ( x, 0)  f ( x) (1 x e).
where p( x)  x ,  ( x)  1/ x 2 . The boundary conditions (2)-(a), (b) will be of the first
or second kind, and A and B will be given constants.
(a)-[ 10 pts.]- Find U ( x) the equilibrium solution of the given initial-boundary value problem.
(b)-[ 5 pts.]- Set v( x, t )  u ( x, t )  U ( x) , and determine the partial differential equation, the
boundary conditions, and the initial condition satisfied by v( x, t ) .
(c)-[10 pts.]-Use the method of separation of variables to solve for v( x, t ) .
See my supplement on eigenvalue problems for Cauchy –Euler equations.
See section 8.1, 8.2, 8.3. Do problem (8.2.1) on page 352, and problem (8.2.6) on page
353. Do problem (8.3.2) and (8.3.4) on page 358-359. Also see section 5.3.1. Do
problem (5.3.9) on page 169. See section 5.4.
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Problem #2:
Consider the following initial-boundary value problem for the 1-dimensional wave equation
(1) utt  c 2uxx (0  x  L, t 0),
(2)-(a)  u (0, t )   ux (0, t )  A(t ), (2)-(b)  u(L, t )   u x (L, t )  B(t ) (t
0),
(3)-(a) u ( x, 0)  0, (3)-(b) ut ( x, 0)  0 (0 x L),
where the boundary conditions (2)-(a), (b) will be of the first or second kind, and where
A(t ) and B(t ) are given functions of t .
(a)-[ 5 pts.]- Find the coefficients a(t ) and b(t ) in the function r ( x, t )  a(t )  b(t ) x
so that  r (0, t )   rx (0, t )  A(t ), (2)-(b)  r (L, t )   rx (L, t )  B(t ) (t 0).
(b)-[ 5 pts.]- Set v( x, t )  u ( x, t )  r ( x, t ) , and determine the partial differential equation,
the boundary conditions, and the initial condition satisfied by v( x, t ) .
(c)-[10 pts.]- Find v( x, t ) by the eigenfunction expansion method.
See section 8.2, pp. 347-352. See section (8.3) pp.354-358. Do problem (8.2.2) and
(8.2.3) on page 352-353. Do problem (8.2.6) on page 353.
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Problem #3:
Consider the following initial-boundary value problem for the 1-dimensional heat
(1)  ( x)ut  ( p( x)ux ) x (0  x  L, t 0),
(2)-(a) u (0, t )  0, (2)-(b) u (L, t )  0 (t 0),
(3) u ( x, 0)  f ( x) (0 x L)
where p( x)   ( x)  eax (a  constant)
(a)-[15 pts.]- Find a (complete) sequence of eigenfunctions n ( x) that satisfy the SturmLiouville equation ( p( x) ') '  ( x)  0 , and the boundary conditions  (0, t )   (L)  0 .
equation:

(b)-[10 pts.]- Determine the coefficients An (t ) in the infinite series
 A (t ) ( x) so that
n 1
n
n

u ( x, t )   An (t )n ( x)
n 1
is the solution of the given initial-boundary value problem.
See my supplement on eigenvalue problems for differential equations of the form
 ''( x)   '( x)   ( x)  0. See section 8.4 on pp. 359-363. Do problems (8.4.1)- (8.4.3)
on page 363. Do problem (8.4.4) on page 363-364. See section (5.4) on pp. 170-172.
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STUDY GUIDE-FINAL EXAM
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Problem #4:
Use the eigenfunction expansion method to solve the following boundary value
problem for Laplace's equation.
u  u xx  u yy   ( x, y ) (0  x  L, 0  y  H )
 u(0, y)   ux (0, y)  A( y),  u(L, y)   ux (L, y)  B( y) (0  y  H ),
 u ( x, 0)   u y ( x, 0)  0, u ( x, H )   u y ( x, H )  0 (0  x  L),
where the boundary conditions (2)-(a), (b) will be of the first or second kind, and
where A( y ) and B( y ) are given functions of y .
(a)-[10 pts.]- Find a (complete) sequence of eigenfunctions n ( y) that satisfy the SturmLiouville equation  ''( y )   ( y )  0, and the boundary conditions
  (0)    '(0)  0,  ( H )    '( H )  0.

(b)-[15 pts.]- Determine the coefficients An ( x) in the infinite series
 A ( x) ( y )
n 1
n
n

series so that u ( x, y )   An ( x)n ( y ) is the solution of the given boundary value problem.
n 1
See section (8.6), pp. 372-377, and do problem (8.6.1) on page 368, and problem (8.6.6)
on page 378.
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Problem #5:
(a)-[10 pts.]-Find the eigenfunctions and an equation for the eigenvalues for the
following Sturm-Liouville problem:
 '' a '   0 (0 x L)
 (0)   '(0)  0, ( L)   '( L)  0
where the boundary conditions will be of the first or second kind, and a is a constant.
(b)-[5 pts.]-Use a graphical argument to find upper and lower bounds for the smallest eigenvalue.

(c)-[10 pts.]- Determine the coefficients An (t ) in the infinite series
 A (t ) ( x) so that
n 1
n
n

u ( x, t )   An (t )n ( x)
n 1
is the solution of the wave equation e utt  (eaxu x ) x (0 x L, t 0) that satisfies the
boundary conditions  u(0, t )   ux (0, t )  0, u( L, t )   ux ( L, t )  0 (t 0), and the
initial conditions u( x,0)  f ( x), ut ( x,0)  g ( x) (0  x  L) .
See my supplement on eigenvalue problems for differential equations of the form
 ''( x)   '( x)   ( x)  0. See section 5.8, pp. 198-first paragraph on page 202.
Do problems (5.8.3) and (5.8.5)-a. and (5.8.8) on ppp. 209-210. Do problem (5.8.6) on
page 210.
ax
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Problem #6:
[25 pts.]-Use the method of separation of variables to find the solution of the circularly
symmetric heat equation
1
urr  ur  ut (0 r 1, t 0)
r
that satisfies (1) the finiteness condition u(0, t )  (t 0) , (2) the boundary condition
a u(a, t )  b ur (a, t )  0 (t 0) (where a or b is zero), and (3) the initial condition
u (r , 0)  f (r ) (0  r  1) .
See section 7.7.9 on pp 313-315. Do problems (7.7.1)-(7.7.2) on page 315. Do problem
(7.7.10)-(7.7.11) on page 317.
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STUDY GUIDE-FINAL EXAM
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Problem #7:
Consider the following boundary value problem for Laplace’s equation:
1
1
(1) u zz  urr  ur  2 u  0 (0     , 0 r  1, 0  z  1),
r
r
(2) a u (r , ,1)  b u z (r , ,1)  f ( r , ) (0     , 0 r  1),
(3) c u (r , , 0)  d u z (r , , 0)  0 (0     , 0
(4) u (0, , z )
r  1),
, u (1, , z )  0 (0     , 0  z  1),
(5) u (r , 0, z )  u (r ,  , z )  0 (0
r  1, 0  z  1)
where the boundary conditions (2) and (3) will be of the 1st or second kind.
(a)-[15 pts.]- Find a countable infinity of product solutions
umn (r , , z )  hmn ( z )mn (r , ) (m, n  1, 2,3,...)
that satisfy equation (1), and conditions (3)-(5).
(b)-[10 pts.]- Show how to solve the given boundary value problem by expressing the
solution as an infinite linear combination of the product solutions found in part (a),
that is, set


u (r , , z )   hmn ( z ) mn (r , ) ,
n 1 m 1
and determine the constants hmn (1) so that the above series satisfies the inhomogeneous
boundary condition (2).
See section 7.9.1-7.9.3 on pp. 326-329. Do problem (7.9.2) on page 335.
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