MATH 412: Midterm 1
Exam time: 75 minutes
Thursday, Feb 6th, 2014
Name: 3 QQMMOW 5
Note: Write your solutions clearly and explain all your steps in order to receive
the full credit. This is a closed book exam. N o textbooks, notebooks, calculators,
no
MATH 412 Fourier Series and PDE- Spring 2010
SOLUTIONS to HOMEWORK 5
Problem 1.
(a): Solve the following Sturm-Liouville problem
(xu ) + u = 0
x
u(1) = u (e) = 0.
1<x<e
(b): Show directly that the eigenfunctions are orthogonal with respect to the
suitable
MATH 412. Partial Dierential Equations and
Fourier Transform
Spring 2016. 203 Electrical Eng West, Tue/Thu 2:30 3:45.
Instructor: Alexei Novikov. OH: Tue/Thu 1:30-2:30, McAllister 233.
Book: W. Strauss, Partial Dierential Equations: An Introduction, 2nd E
1
Introduction
1.1
Notation and Basic Denitions
If u is a function of variables (x1 , x2 , . . . , xn ), then its partial derivative with
respect to xi is denoted by
xi u
or
u
xi
or
ux i .
To denote higher-order derivatives we use of a notion of a multi-i
7
Separation of Variables
Chapter 5, An Introduction to Partial Dierential Equations, Pichover and
Rubinstein
In this section we introduce the technique, called the method of separations
of variables, for solving initial boundary value-problems.
7.1
Heat
4.1
Conservation Laws
We consider the initial value problem for equations describing scalar conservation laws,
ut + (f (u)x = 0 on R [0, )
(4.1)
u(x, 0) = (x)
where f is of class c1 . The name conservation law refers to the following
fact. If u(x, t) is a
5.1
Second-Order linear PDE
Consider a second-order linear PDE
L[u] = auxx + 2buxy + cuyy + dux + euy + f u = g,
(x, y ) U
(5.1)
for an unknown function u of two variables x and y . The functions a, b and
c are assumed to be of class C 1 and satisfying a2
Elliptic Equations
Laplace equation on bounded domains
Sections 7.7.2, 7.7.3, An Introduction to Partial Dierential Equations, Pinchover and Rubinstein
10.2
Circular Domains
We study the two-dimensional Laplace equation
u = uxx + uyy = 0
(x, y ) Ba
(10.18
6
Wave Equation
Pinchover and Rubinstein, Chapter 4.
We consider the homogeneous wave equation in one-dimension,
utt c2 uxx = 0,
a < x < b , t > 0
(6.1)
To nd the general solution of (6.1), we can proceed as follows.
We introduce the new coordinates whic
10
Elliptic equations
Sections 7.1, 7.2, 7.3, 7.7.1, An Introduction to Partial Dierential Equations, Pinchover and Rubinstein
We consider the two-dimensional Laplace equation on the domain D ,
u = 0,
(x, y ) D.
u = F,
(x, y ) D
More general equation
is c
9.3
Separation of variables for nonhomogeneous equations
Section 5.4 and Section 6.5, An Introduction to Partial Dierential Equations, Pinchover and Rubinstein
The method of separation of variables can be used to solve nonhomogeneous
equations. We only co
9
Sturm-Liouville Problems and Eigenfunction expansion
Chapter 6, An Introduction to Partial Dierential Equations, Pinchover and
Rubinstein
9.1
Inner Product Spaces
Let V be a (real) vector space. Then a function , : V V R is called
the inner product or s
8
Fourier Series
Our aim is to show that under reasonable assumptions a given 2 -periodic
function f can be represented as convergent series
a0
(an cos nx + bn sin nx) .
+
2
n=1
f (x) =
(8.1)
By denition, the convergence of the series means that the seque
1
First-Order Equations
In this section we describe the technique for solving rst-order quaslinear
equation
a(x, y, u)ux + b(x, y, u)uy = c(x, y, u).
1.1
Semilinear equations
We rst consider semilinear equations
a(x, y )ux + b(x, y )uy = c(x, y, u),
(1)
w
MATH 412 Fourier Series and PDESpring 2010
HOMEWORK 5 Due: Thursday, April 8.
Submit solutions to all of the problems. Each problem is worth the same number of points. Collaboration is allowed, but you need to turn in individually
written solutions. The p
MATH 412 Fourier Series and PDE- Spring 2010 SOLUTIONS to HOMEWORK 2
Problem 1. Solve the following Cauchy problem, u2 + uy + u = 0 x u(x, 0) = x. Solution: Here F (x, y, z, p.q ) = p + p + z and the initial curve is given by (x0 (s), y0 (s), u0 (s) = (s,
3.1
Quasillinear equations
Given an initial curve (s) = (s), u0 (s) = (x0 (s), y0 (s), u0 (s) in R3 we
are interested in nding a solution of the initial value problem for the rstorder qusillinear equation,
a(x, y, u)ux + b(x, y, u)uy = c(x, y, u)
u| = u0
Math412Extra Problems
Problem 1. Solve the following initial vale problems:
x2 ux + y 2 uy = u2
u(x, 2x) = 1
Solution: The characteristic equations are
x = x2
y = y2
z = z2
and the initial conditions are
x(0, s) = s,
y (0, s) = 2s,
z (0, s) = 1.
The solut
MATH 412: Fourier Series and Partial Differential Equations
Spring 2014
Homework Assignments
Set 1: due Jan 23.
Sec 1.1: 2,3
Sec 1.2: 1,2,3,5,9
Set 2: due Jan 30.
Sec 1.2: 8, 11
Sec 1.3: 6
Sec 1.5: 1,5,6
Set 3: due Jan Feb 13 (but strongly encouraged to w
MATH 412: Fourier Series and Partial Differential Equations
Spring 2014
Homework Assignments: Solutions
Set 1: due Jan 23.
Sec 1.1: 2
a. Linear
b. Nonlinear, due to uuy term.
c. Nonlinear, due to u2 .
y
d. Not linear, due to +1 constant: indeed, L(0) is n
MATH 412: Fourier Series and Partial Differential Equations
Spring 2014
Homework Assignments: Solutions
Set 2: due Jan 30.
Sec 1.2: 8
Along the characteristic lines cfw_y = bx/a + y0 , we get
d
b
c
u(x, y (x) = ux + uy = u(x, y (x).
dx
a
a
This gives
u(x,
MATH 412 Fourier Series and PDE- Spring 2010
HOMEWORK 3 Due: Tuesday, March 2.
Submit solutions to all of the problems. Each problem is worth the same number
of points. Three randomly selected problems will be graded. Collaboration is
allowed, but you nee
Math412Extra Problems
Problem 1. Solve the following initial vale problems:
(a)
y ux + xuy = 0
u = sin x on the circle x2 + y 2 = 1.
(b)
xux uuy = y
u(1, y ) = y.
Is the solution unique? What is the maximal domain where it is dened?
(c)
ux + u2 uy = 0
u(x
MATH 412 Fourier Series and PDE- Spring 2010
HOMEWORK 1 Due: Thursday, February 4.
Submit solutions to all of the problems. Each problem is worth the same number of points. Three randomly selected problems will be graded. Collaboration
is allowed, but you
MATH 412 Fourier Series and PDESpring 2010
Final Exam Review Problems
Below is the list of problems that could be helpful in your preparation to the
exam. All problems (except problem 3) are from the book An introduction to
partial dierential equations by
MATH 412 Fourier Series and PDE- Spring 2010
SOLUTIONS to HOMEWORK 6
Problem 1. Solve the following problem
ut uxx = 1 + x cos t
u (0, t) = ux (1, t) = sin t
x
u(x, 0) = 1 + cos(2 x)
0 < x < 1, t > 0
t0
0 < x < 1, t > 0.
Solution: We look for the solution
MATH 412 Fourier Series and PDESpring 2010
HOMEWORK 4 Due: Thursday, March 25.
Submit solutions to all of the problems. Each problem is worth the same number
of points. Collaboration is allowed, but you need to turn in individually written
solutions. Most
MATH 412 Fourier Series and PDESpring 2010
HOMEWORK 6 Due: Thursday, April 29.
Submit solutions to all of the problems. Each problem is worth the same number of points. Collaboration is allowed, but you need to turn in individually
written solutions. The