M597K: Solution to Homework Assignment 14
(Last One)
Date: Dec 9, Monday; Due by Monday Dec. 16
1. Use Rayleigh quotient (Section 6.12, Conclusion No. 6) to show that any eigenvalue must be positive ( > 0) for Bessels equation
(ru )
m2
ru
+ ru = 0,
0 <
M597K: Solutions to Homework Assignment
13
Date: Dec. 16, Monday
1. Use change of variables to reduce
2
u
t
= k u + Q(t, x),
x2
0 < x < L,
u(0, x) = g(x),
u
x (t, 0)
u
x (t, L)
(1)
= A(t),
= B (t)
to a problem with homogeneous boundary condition.
Solution
M597K: Solution to Homework Assignment 12
Date: Nov. 25, Monday; Due Wed. Dec. 4.
1. Use separation of variables to derive a solution formula for
2u
x2
+
2u
y 2
= 0,
0 < x < L, 0 < y < H
u(0, y ) = 0,
(1)
u(L, y ) = 0,
u(x, 0) = 0,
u(x, H ) = g(x).
Soluti
M597K: Solution to Homework Assignment 11
Date: Nov. 18, Monday; Due Wed. Nov. 27.
1. Find a solution to
2u 2u
+2
x2
y
u
t
au = 0,
where a is a constant, with initial condition
u(0, x, y ) = g(x, y ).
Hint: Show that v = eat u satises the standard heat e
M597K: Solution to Homework Assignment 10
Date: Nov. 11, Monday; Due Wed. Nov. 20.
1. Use the method of characteristics to derive a solution formula to
u
u
+a
+ cu = 0,
t
x
u(0, x) = g(x),
where a and c are constants, t > 0, and x IR 1 . (Hint: Derive an
M597K: Solution to Homework Assignment 9
1. Solve the initial value problem for a rst-order linear homogeneous equation
dx
(sin t) x = 0, (t > 0);
dt
x(0) = 1.
Solution.
dx
= (sin t) dt
x
t dx
t
sin t dt
=
0x
0
ln x(t) ln x(0) = cos t|t = 1 cos t
0
using
M597K: Solution to Homework Assignment 8
1. Follow the proof of Property b to prove
(f g) () =
2f () g ().
Solution.
(f g) () =
=
ix
( f (x y )g(y )dx
e
i(xy ) g (y )eiy dy ) dx
(f (x y )e
1
2
1
2
(1)
Let x y = z
(f g) () =
=
iz )dz g (y )eiy dy
(f (
M597K: Solution to Homework Assignment 7
The following problems are on the specied pages of the text book by Keener
(2nd Edition, i.e., revised and updated version)
Problems 3 and 4 of Section 2.1 on p.94;
Problem 1 of Section 3.1 on p.128;
Problem 1 of S
M597K: Solutioin to Homework Assignment 6
Date: October 2, Wednesday, 2002. Due Friday. Oct. 11.
1. Find the numerical value of each of the following and express it in the form a + bi
where a and b are real numbers.
(a) i(1 + i)(2 + i),
(b) (1 i)2 + (3 2i
M597K: Solutioin to Homework 5
Date: Friday, October 4.
1. Recall that the product of two n n matrices A = (aij ) and B = (bij ) is dened
as the matrix AB = (cij ) where
n
cij =
(i, j = 1, 2, , n).
aik bkj
k =1
Thus show that
u1
x
u2
x
u3
x
u1
y
u2
y
u3
y
M597K: Solution to Homework 4
Date: Friday Sept 27, 2002
1. Given a scalar (x1 , x2 , x3 ), does the gradient
= (x1 , x2 , x3 ) satisfy
the law of coordinate transformation for rst-order tensors? That is,
xi (x1 , x2 , x3 ) = i k xk (x1 , x2 , x3 )?
Here
M597K: Solution to Homework 3
Date: Sept. 20, Friday
Solutions 1-2: Omitted.
3. (Summation convention) Expand the terms Ai B k Ci and aij bj . Is there a summation in ai + bi ?
Solution: (20 points) Ai B k Ci = B k (A1 C1 + A2 C2 + A3 C3 ).
aij bj = ai1 b
M597K: Solution to Homework Assignment 2
Date: Friday, Sept. 13, 2002.
1. Find the derivative of the vector A(t) = (cos t, sin t, 2t). Draw the graph of A(t)
with A (t).
The derivative is A (t) = ( sin t, cos t, 2). The graph is
Solution. (10 points)
in F
M597K: Solution to Homework Assignment 1
Date: Sept 6, 2002.
1. Let e1 , e2 , e3 be a basis (may or may not be orthonormalized). Let A be a
vector. Is it always true that
Proje1 A + Proje2 A + Proje3 A = A?
Solution (10 points). No, it is not always true.