Mathematics 4582			Final				December, 1997
NAME:_________________________________

Instructions: Write the answers where indicated (or points will be taken off).  If you do not understand what is being asked for, ask Prof. Harrell for a clarification.

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NAME________________________

THERE MAY BE HELPFUL INFORMATION ON THE PAGES AT THE END OF THE TEST!

1.  

Solve the wave equation

	utt = 9 uxx, 0 < x < ∞, 0 < t < ∞

with boundary condition

	ux(t,0) = 0

and initial conditions

	u(0,x) = exp(-x2)

	ut(0,x) = sin(x)

ANSWER:

	u(t,x) = _____________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________ 
________________________________________________________________

NAME________________________

THERE MAY BE HELPFUL INFORMATION ON THE PAGES AT THE END OF THE TEST!

2.  Suppose that a thin disk of radius 1 is insulated on its faces while the round edge is held at temperature 0.  The thermal properties of the disk are such that the heat equation reads:

	ut = 100 —2 u

At time 0, the temperature distribution is:

	u(0,r,q) = r2 sin(2q)

What is the temperature at time t and position r, q ?

ANSWER:

u(t,r,q) = __________________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________ 
________________________________________________________________

NAME________________________

THERE MAY BE HELPFUL INFORMATION ON THE PAGES AT THE END OF THE TEST!

3.  Solve Laplace's equation (for the electric potential in the absence of charge) on the rectangle
	-1 < x < 1
	  0 < y < 1

with the following boundary conditions:

	u(x,0)   = 0, 
	u(x,1)   = 1, 
	u(-1,y)  = 0, 
	ux(1,y) = 1. 

What is the potential inside the rectangle?

ANSWER:

u(x,y) = __________________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________ 
________________________________________________________________

Possibly helpful information:

The Laplace operator in various coordinate systems:

Bessel functions
Two independent solutions of 
	r2 R'' + r R' + (µr2 - m2) R = 0 

are J±m(µ1/2 r), when m= 1,2,...; For small s, Jm(s) ~ C sm.  When m=0, two independent solutions are J0(µ1/2 r) and Y0(µ1/2 r); J0 is bounded at r=0, but Y0 is not.
The for each fixed integer m> 0, the set of functions Jm(jmnr/A) is complete for 0 < r < A, with the orthogonality relationship

Some help from Waterloo, Ontario:

> int(sin(a*x) * (sin(b*x))^2, x);

                cos(a x)       cos((a + 2 b) x)       cos((a - 2 b) x)
          - 1/2 -------- + 1/4 ---------------- + 1/4 ----------------
                    a               a + 2 b                a - 2 b

> int(cos(a*x) * (sin(b*x))^2, x);

               sin(a x)       sin((a - 2 b) x)       sin((a + 2 b) x)
           1/2 -------- - 1/4 ---------------- - 1/4 ----------------
                   a               a - 2 b                a + 2 b

> int(sin(2*b*x) * (sin(b*x))^2, x);

                             cos(2 b x)        cos(4 b x)
                       - 1/4 ---------- + 1/16 ----------
                                  b                 b

> int(cos(2*b*x) * (sin(b*x))^2, x);

                        sin(2 b x)        sin(4 b x)
                    1/4 ---------- - 1/16 ---------- - 1/4 x
                             b                 b

> int(sin(a*x) * x, x);

                           sin(a x) - a x cos(a x)
                           -----------------------
                                       2
                                      a

> int(cos(a*x) * x, x);
                           cos(a x) + sin(a x) a x
                           -----------------------
                                       2
                                      a

> int(sin(a*x) * x^2, x);
                  2  2
               - a  x  cos(a x) + 2 cos(a x) + 2 sin(a x) a x
               ----------------------------------------------
                                      3
                                     a

> int(cos(a*x) * x^2, x);
                          2  2
                sin(a x) a  x  - 2 sin(a x) + 2 a x cos(a x)
                --------------------------------------------
                                      3
                                     a