Quiz 6
Math 309E. March 10, 2014
Solve the following BVP:
utt = uxx , u|t=0 = 0, ut |t=0 =
2, 1 < x < 2;
0, else,
ux |x=0 = ux |x=3 = 0.
Solution. Try to nd the solution in the form u(t, x) = T (t)X(x) which will satisfy the wave
equation and the boundary
Homework 6, due February 26, 2014
Problem 1. Solve the following boundary value problem: y + y = 0, y (0) = 0, y (L) = 0.
Write the solution in full detail, discussing all cases.
Solution. In each of the following three cases, we rst nd the general soluti
Homework 4 Solution, due February 5, 2014
For Problems 1-4,
(i) nd the type of the phase portrait and sketch it (for the case of a node and a saddle point, indicate
the two eigenvectors on the picture and nd them);
(ii) indicate whether it is stable, unst
Homework 5 Solution, due February 19, 2014
In Problems 1-3, decompose the function into the Fourier series on the given interval. Always write the
answer in the sigma notation. If required, also write a few rst terms of the series explicitly.
Problem 1. f
Homework 1 Solution, due January 15, 2014
For Problems 1-6,
(i) nd eigenvalues and eigenvectors of the following matrices;
(ii) indicate the algebraic and geometric multiplicity of each eigenvalue;
(iii) indicate whether each of these matrices is defectiv
Homework 2 Solution, due January 22, 2014
For Problems 1-6, solve the system x (t) = Ax(t) for the following matrices A:
Problem 1. A =
1 1
0 1
Solution. The characteristic polynomial: 2 2 + 1. Its roots: = 1. Eigenvectors corresponding to
= 1:
v1 + v2 =
Lecture 18. Heat Equation II. February 24, 2014
Let us solve
ut = a2 uxx , u|t=0 = f (x), ux |x=0 = ux |x=L = 0.
Here, a, L > 0 are some constants.
Try an example:
1
ut = uxx , u|t=0 = cos2 (2x), ux |x=0 = ux |x=1 = 0.
2
Try to nd a solution of this equat
Lecture 17. Heat Equation I. February 21, 2014
A heat equation is an equation of the form
ut = a2 uxx ,
where a > 0 is a constant, and u(t, x) is a function of two variables. The notation ut means
the derivative of u with respect to t. The notation uxx me
Lecture 19. Heat Equation III. February 26, 2014
Dierent types of trig series on [0, L]:
1. Cosine Series.
L
nx
2
a0
+
an cos
, an =
f (x) =
2
L
L
n=1
f (x) cos
0
nx
dx, n 0.
L
2. Sine Series.
f (x) =
bn sin
n=1
L
2
nx
, bn =
L
L
f (x) sin
0
nx
dx, n 1.
L
Lecture 20. Wave Equation I. February 28, 2014
Wave Equation: utt = a2 uxx with initial conditions
u|t=0 = f (x), ut |t=0 = g(x)
and boundary conditions
Case 1. u|x=0 = u|x=L = 0.
Case 2. ux |x=0 = ux |x=L = 0.
Try an example of Case 1. Let
utt = uxx , u|
Lecture 12. Nonhomogeneous Systems II. February 3, 2013
There is another way of solving nonhomogeneous systems:
x (t) = Ax +
f (t)
2 1
, A=
g(t)
1 2
Diagonalize the matrix A: Eigenvalue problem:
1
1
; = 3, v2 =
1
1
= 1, v1 =
So
Av1 = v1 , Av2 = v2 AS = S
Homework 3 Solution, due January 29, 2014
For Problems 1-2, solve the system x (t) = Ax(t) for the following matrices A:
Problem 1. A =
2 1
4 2
Solution. The characteristic polynomial: 2 4 + 8. Roots (eigenvalues): = 2 2i. Eigenvector
corresponding to = 2
Lecture 22. Wave Equation III. March 5, 2014
Consider an example of a boundary value problem for the wave equation:
utt = 9uxx ,
with boundary conditions
u|x=0 = u|x=2 = 0,
and initial conditions
u|t=0 = sin(6x), ut |t=0 = sin(2x) sin(4x).
Separation of v
Lecture 24. Laplace Equation I. March 10, 2014
Consider the BVP for the Laplace equation u = uxx + uyy = 0 in the disc: x2 + y 2 a2 ,
where a is the radius of this disc. The boundary condition is u = f on the circle which is the
boundary of the disc. Here
Quiz 4 Solution
Math 309E. February 21, 2014
Decompose the function f (x) = (1/3,1) into the Fourier series on [1, 1]. Write the answer in
sigma notation.
Solution. The Fourier series has the form
a0
f (x) =
+
(an cos nx + bn sin nx).
2
n=1
Here,
1
1
f (x
Quiz 5 Solution
Math 309E. February 28, 2014
Solve the following initial-boundary value problem for the heat equation:
ut = 6uxx , u|t=0 = 3 cos2 (2x), ux |x=0 = ux |x=/2 = 0.
Solution. Try u(t, x) = T (t)X(x); then we have:
X (x)
T (t)
=
= ,
6T (t)
X(x)
Quiz 2 Solution
Math 309E. January 24, 2014
Find a solution to the initial value problem
x1 (t) = 3x1 (t) 2x2 (t), x1 (0) = 2
x2 (t) = 2x2 (t) + 3x2 (t), x2 (0) = 1.
Solution. The matrix of the system is
A=
3 2
2 3
Its characteristic polynomial is 2 6 + 5
MATH 309E
Midterm
February 7, 2014
Student ID #
Name
Your exam should consist of this cover sheet, followed by 4 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless otherwise in
Quiz 1 Solution
Math 309E. January 17, 2014
Find all eigenvalues and eigenvectors of the matrix A. Indicate algebraic and geometric multiplicity of each eigenvalue. Is this matrix defective or not? If it is not defective, diagonalize
it.
5 1
A=
1 7
Soluti
MATH 309E
Midterm Solution
February 7, 2014
Student ID #
Name
Your exam should consist of this cover sheet, followed by 4 problems. Check that you have
a complete exam.
Unless otherwise indicated, show all your work and justify your answers.
Unless oth
Lecture 21. Wave Equation II. March 3, 2014
Now, consider the wave equation
utt = a2 uxx
with initial conditions
u|t=0 = f (x), ut |t=0 = g(x),
and boundary conditions
ux |x=0 = 0, ux |x=L = 0.
Separation of variables: let u(t, x) = X(t)T (t), we have:
T
Lecture 25. Laplace Equation II. March 12, 2014
Consider the BVP for the Laplace equation u = uxx + uyy = 0 in the disc: x2 + y 2 a2 ,
where a is the radius of this disc. The boundary condition is u = f on the circle which is the
boundary of the disc. We
Practice Problems for the Final. Math 309, Winter 2014
Problem 1. Solve the system
x (t) =
2 2
e4t
x(t) +
tet
1 1
(try to solve it both using diagonalization and undetermined coecients).
Problem 2. Solve the eigenvalue problem for
1 0 1
A = 0 1 0
1 1 1
Pr
Lecture 15. Fourier Series III. February 14, 2014
Fourier Series on dierent intervals. For [0, 2L] or [L, L] instead of [, ], we can also
consider Fourier series:
nx
nx
a0
+
an cos
+ bn sin
.
f (x) =
2
L
L
n=1
Here,
L
1
an :=
L
nx
1
f (x) cos
dx, n 0; bn
Lecture 16. Boundary Value Problems. February 19, 2014
Problem 1. Consider the dierential equation y + y = 0, with boundary value conditions:
y(0) = y() = 0. These are dierent from initial value conditions, like y(0) = y (0) = 0. Solve
it:
y = C1 cos t +
Autumn 2012, Math 309 E
Sample Quiz 8
1) Find a suitable harmonic function on D and use the Poisson integral
1
formula with r = 2 and = to calculate the value of
2
2
0
sin(2t)
dt.
1 4 sin(t)
5
Autumn 2012, Math 309 E
Quiz 4
1. Find the positive eigenvalues and the corresponding eigenfunctions
associated to the boundary value problem y + y = 0 with y (0) = 0 and
y (L) = 0. Show all work. You do not need to show why there are no nonpositive eigen
Autumn 2012, Math 309 E
Quiz 5
1) Find the Fourier cosine series for f (x) = x2 on x . You may
2x cos(ax) a2 x2 2
need that x2 cos(ax)dx =
+
sin(ax) + C .
a2
a3
2) Use problem 1 to nd the value of the innite series
n=1
(1)n
.
n2
Autumn 2012, Math 309 E
Quiz 6
1) Find the steady-state solution to the heat equation: 4uxx = ut for
0 < x < 6 and t > 0 satisfying u(0, t) = 1, u(6, t) = 2, and u(x, 0) = x2 .
2) If w(x, t) is the transient part of u for the above equation, write down
th