Exam 1, MAP 6506, Spring12, Due March 19, 2012, (by 3 pm in class)
Consider the Derichlet and Neumann problems for the two-dimensional Laplace equation.
The same notations are adopted as in lectures. An open bounded region G is bounded by a
curve S that h
Exam 1, MAP 4341/5345, Spring12
1. Determine the equilibrium temperature distribution for one-dimensional rod with constant
thermal properties:
ut(x, t) = kuxx(x, t),
u(x, 0) = x(L x),
ux (0, t) u(0, t) + T = 0,
ux (L, t) = a
where T and a are constants,
Exam 1, MAP 4341/5345, Spring12
1. Show your work! Write your name on every piece of paper you turn in!
2. Write your alias name on the back of the exam sheet.
3. You may use appropriate Fourier series for the solutions satisfying the zero boundary condit
Exam 3, MAP 4341/5345, Spring12
1. Show your work! Write your name on every piece of paper you turn in!
2. You may use appropriate Fourier series for the solutions satisfying the boundary conditions (as in
the homework) without deriving them.
1. Solve the
Homework 1, MAP 6506, Spring12, Due March 19, 2012, (by 3 pm in class)
1. Consider an integral equation with Volterra kernel:
x
u (x ) =
a
K (x, y )u(y )dy + f (x)
where K (x, y ) is continuous in the closed triangle a y x b and f is continuous on [a, b]
Homework 1, MAP 4341/5345, Spring12
1. Solve the initial value boundary problem
ut(x, t) = kuxx (x, t),
u(x, 0) = T,
u(0, t) = 0,
ux (L, t) = 0
where T is a constant.
Answer:
2
an ekn t sin(n x),
u(x, t) =
n =
n
n
,
2L
an =
4T
[1 cos(n L)],
n
n = 1, 3, 5,
Homework 3, MAP 4341/5345, Spring12
1. Let
A=
6 2
2 3
(see also the textbook Example on p. 186 and your notes). Use the orthogonal basis of the
eigenvectors of the matrix A to solve each of the following initial value problems:
(i)
dx
0
4
= Ax + f , f =
,