Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod*
At Georgia Tech, this was used as the final exam after a short term (less than 10 weeks). Students were allowed 2 hours, 50 minutes, which was adequate. Few students got problem 3 completely correct.
The announced standard for the test was:
A = 2 problems completely solved, a correct idea on the third. B = 1 problem completely solved, correct ideas on the other two. C = good ideas on the all problems, but no complete solutions.
THERE MAY BE HELPFUL INFORMATION ON THE PAGES AT THE END OF THE TEST!
1.
a) Solve the following problem:
PDE ut = 4 uxx, for 0 < t, 0 < x < 1
BC ux(t,0) = 0, ux(t,1) = 0, for 0 < t
IC u(0,x) = 2 x - x2, for 0 < x < 1
ANSWER:
u(t,x) = _______________________________________
b) Find the maximum value of u(t,x) for 0 <= t <= 1, 0 <= x <= 1:
ANSWER:
The maximum temperature occurs at x = _____, t = _____
The maximum temperature is umax = _________________
THERE MAY BE HELPFUL INFORMATION ON THE PAGES AT THE END OF THE TEST!
2. Some background information:
There is a complete, orthonormal set of functions denoted \phin(x), for -infinity < x < infinity, which are eigenfunctions for the ordinary differential equation
- \phin'' + x2 \phin = (2n+1) \phin, n = 0, 1, 2, ....
You may use the notation \phi n in the answer to this problem.
Consider the following PDE:
PDE utt = grad2 u - x2 u, for 0 < t, 0 < x < \pi
BC u(t,x,0) = u(t,x, \pi ) = 0, for 0 < t
Find the normal mode with the lowest frequency of vibration (include the time dependence):
ANSWER:
u(t,x,y) = __________________________________________________
Find the general solution:
ANSWER:
u(t,x,y) = __________________________________________________
3. A slice of pizza is shaped like a sector in cylindrical coordinates,
0 < r < 20 cm,
0 < \theta < \pi /3 radians
0 < z < 1 cm.
It has come to thermal equilibrium while sitting on a student's computer monitor, so that the temperature on its surface is
u(r, \theta , 0) = 30
u(r, \theta , 1) = 20
u(r,0,z) = u(r, \pi /3,z) = 30 - 10 z
u(20, \theta ,z) = 30 - 10 z - 5
This is a low quality pizza consisting of a homogeneous material (independent of position)
a) The partial differential equation for a homogeneous material at thermal equilibrium is:
_______________________________________
b) Answer the following questions.
Are there useful simplifications involving the boundary conditions? If so, what are they?
Be specific and put the answer here:_____________________________________
Are there useful separated solutions? If so, write the specific ordinary differential equations that the separated solutions satisfy below. Include boundary conditions.
____________________________________________________
____________________________________________________
____________________________________________________
____________________________________________________
c) Solve the differential equation with the given boundary conditions.
The Laplace operator in various coordinate systems:
Bessel functions
Two independent solutions of
r2 R'' + r R' + (\mu r2 - m2) R = 0
are J+/-m(\mu1/2 r), when m= 1,2,...; For small s, Jm(s) ~ C sm. When m=0, two independent solutions are J0(u1/2 r) and Y0(u1/2 r); J0 is bounded at r=0, but Y0 is not.
For each fixed integer m> 0, the set of functions Jm(jmn r/A) is complete for 0 < r < A, with the orthogonality relationship
An eigenvalue problem. There is a complete, orthonormal set of functions denoted \phi n(x), for -infinity < x < infinity, which are eigenfunctions for the ordinary differential equation
- \phin'' + x2 \phin = (2n+1) \phin, n = 0, 1, 2, ....
In fact,
\phin = exp(-x2/2) hn(x),
where hn is a polynomial in x of degree n, and that \phin is even when n is even, and odd when n is odd.
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
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