Study Guide Test #3- 449/549
THE FOLLOWING IS A COMPLETE LIST OF THE PROBLEMS THAT
WILL APPEAR ON TEST #3 (Final Exam).
ON THE DAY OF THE EXAM YOU WILL BE REQUIRED TO DO ANY
FIVE OF THE ROBLEMS LISTED BELOW.
YOU WLL HAVE THE OPTION OF DOING UP TO FIVE ADDITIONAL
PROBLEMS FOR EXTRA CREDIT. EACH EXTRA CREDIT PROBLEM WILL
BE WORTH ONE HALF THE CREDIT OF A REQUIRED PROBLEM.
Problem #1:
(a) - Determine that the 2nd order partial differential equation
a( x, y ) u xx  2b( x, y ) u xy  c( x, y ) u yy  d ( x, y )u x  e( x, y )u y  f ( x, y )u  0
is either parabolic, hyperbolic or elliptic on a given region in R 2 where the
coefficients in the differential equation are given continuous functions of x and y .
(b) - Find a (characteristic) coordinate transformation that reduces the given p.d.e. to
an appropriate canonical form.
(c) - Use the result of part (b) to calculate the coefficients of the second derive terms
in the canonical form. That is if the canonical form of the given p.d.e. is
~
~
~
a~( , ) u~  2b ( , ) u~  c~( , ) u~  d ( , )u~  e~( , )u~  f ( , )u~  0 ,
~
calculate a~ , b and c~ . Recall that if    ( x, y ),   ( x, y ) is a coordinate
~
transformation then a~  ( )T A , b  ( )T A , c~  ( )T A where
 a b
 is the matrix of the principal part of the differential operator given above.
A  
b c
See Text #1: Problem 7.7 (page 143)
________________________________________________________________________
Problem #2:
Do problem # 13.6 0n page 244 in Text #1.
________________________________________________________________________
Problem #3:
(a)-Consider the Dirichlet problem on the exterior region   R 2  B (0,1)  {( x, y) : r  1}
where B(0,1)  {( x, y ) : r  1} .Use inversion with respect to the
circle S (0,1)  {( x, y ) : r  1} to derive Green’s function for  from the Green’s function
for B (0,1) . The formula for this Green’s function is given in problem (10.4) ( p. 231 of
Text #1).
(b)-Use the result of part (a), and the representation formula (9.11) (p. 224 of Text #1) to
derive the Poisson integral (12.15) (p. 239 of Text #1).
________________________________________________________________________
CONTINUED ON NEXT PAGE
Study Guide Test #3- 449/549
Problem #4:
Derive an integral representation of the solution of the Dirichlet problem for the upper half of the xyz
plane, i.e the region   {( x, y, z ) : z  0} . Note that   {( x, y, z ) : z  0} . Use formula (13.5) (p. 241
of Text #1), and formula (13.4) (p.240 of Text #1).
_______________________________________________________________________________
Problem #5:
(a)-Show that any coordinate transformation    ( x, y ) ,   ( x, y ) from xy space to ξη space
that satisfies the Cauchy-Reimann equations  x ( x, y )   y ( x, y ),  y ( x, y )   x ( x, y ) preserves
Laplace’s Equation.
(b) Show that the inverse x  X ( , ) , y  Y ( , ) of the coordinate transformation of part (a)
satisfies the Cauchy-Reimann equations, i.e. X  ( , )   Y ( , ), Y ( , )  X  ( , ) .
______________________________________________________________________________
Problem #6: 1 Dimensional Wave Equation-(Fourier Transforms)

(a) Show that the function u ( x, y, t )   e  sin(  ( x  t )) d satisfies the wave equation
0
u xx  utt  0 on R by differentiating under the integral sign.
(b) Calculate u ( x, y , t ) , and show by direct substitution that this function satisfies the
wave equation u xx  utt  0 .
_____________________________________________________________________
Problem #7: 1 Dimensional Wave Equation- (Separation of Variables )
Let u ( x, t ) be the solution of the following initial-boundary value problem:
u xx  utt  0 (0  x  1, t  0)
2
u(0, t )  0, u x (1, t )  0 (t  0) ,
u( x,0)  x(1  x), ut ( x,0)  0 (0  x  1) .
(a)-Show by direct substitution that the functions
un ( x, t )  sin n x cos n t (n  1,2,3,...)
 ( 2 n  1 )π 
where λn  
 satisfy u xx  utt  0 on the region (0  x  1, t  0) .
2


(b)-Show that un ( x, t )  sin n x cos n t (n  1,2,3,...) satisfies the boundary conditions
2
u(0, t )  0, u x (1, t )  0 (t  0) .
(c)-The solution of the given initial-boundary value problem can be expressed as
u( x, t )  n1 an sin n x cos n t

if a1 , a2 , a3 ,... are chosen so that u( x,0)  x(1  x)  n1 an sin n x . Calculate

a1 , a2 , a3 ,... .
See Chapter 8, Section 8
_______________________________________________________________________
CONTINUED ON NEXT PAGE
Study Guide Test #3- 449/549
Problem #8: 2 Dimensional Wave Equation- (Separation of Variables )
(a)-Show by direct substitution that the functions
umn ( x, y, t )  sin mx cos ny sin mn t (m  1, 2, 3, ..., n  0,1, 2, 3,...)
where λmn  ( m 2  n 2 )π 2 , satisfy the wave equation u xx  u yy  utt  0 on the region
(0  x  1, 0  y  1, t  0) .
(b)-Show that umn ( x, y, t )  sin mx cos ny sin mn t (m  1, 2, 3, ..., n  0,1, 2, 3,...)
satisfies the boundary conditions u (0, y, t )  0, u (1, y, t )  0 (0  y  1, t  0) ,
u y ( x , 0 , t )  0 , u y ( x , 1, t )  0 ( 0  x  1, t  0 ) .
(c)-The solution u ( x, y , t ) of the following initial-boundary value problem:
u xx  u yy  utt  0 (0  x  1, 0  y  1, t  0)
u (0, y, t )  0, u (1, y, t )  0 (0  y  1, t  0) ,
u y ( x , 0 ,t )  0 , u y ( x , 1, t )  0 ( 0  x  1, t  0 ) ,
u( x, y,0)  0, ut ( x, y,0)  xy (0  x  1, 0  y  1) .
can be expressed as
 


n 0
m1
amn sin mx cos ny sin mn t (m  1, 2, 3, ..., n  0,1, 2, 3,...)
if the amn (m  1, 2, 3, ..., n  0,1, 2, 3, ...) are chosen so that
ut ( x, y,0)  xy  n0 m1 mn amn sin mx cos ny . Calculate amn (m  1, 2, 3, ..., n  0,1, 2, 3, ...) .


See Chapter 8, section 9
________________________________________________________________________
Problem #9:
Use the method of characteristics to solve the following Cauchy problem:
a ( x, y )u x  b( x, y ) u y  c( x, y )u  d ( x, y ), u  u ( x, y ) , u ( x 0 ( ), y0 ( ))  u0 ( ) .
where the coefficients a ( x, y ), b( x, y ), c( x, y ), d ( x, y ) , the initial curve
x  x0 ( ), y  y0 ( ) , and the initial function u0 ( ) are given.
________________________________________________________________________
Problem #10:
Use the method of characteristics to solve the Cauchy problem
aux  b u y  c( x, y)u 2  0, u  u( x, y) , u ( x, y( x))  u0 ( x) .
where the constant coefficients a and b , the coefficient c ( x, y ) , the initial
function u0 ( x) , and the initial curve y  y (x ) are given.
________________________________________________________________________