Math 124A November 04, 2010
Viktor Grigoryan
12
Heat conduction on the half-line
In previous lectures we completely solved the initial value problem for the heat equation on the whole line, i.e. in the absence of boundaries. Next, we turn to problems with
Homework 1
Solutions
1. (#1.1.2 in Strauss) Which of the following operators are linear?
(a) Lu = ux + xuy
(b) Lu = ux + uuy
(c) Lu = ux + u2
y
(d) Lu = ux + uy + 1
(e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u
Solution:
(a) Linear.
(b) Nonlinear the
3.1
DIFFUSION ON THE HALF-LINE
Solutions prepared by Jon Tjun Seng Lo Kim Lin, TA Math 124A
MATH 124A Solution Key HW 05
3.1
DIFFUSION ON THE HALF-LINE
1. Solve u t = ku x x ; u(x, 0) = ex ; u(0, t) = 0 on the half-line 0 < x < .
SOLUTION. By the method o
Math 124a, Winter 16
Midterm Solutions
Exercise 1 Solve ux + yuy = u with the following two conditions respectively:
(a) u(0,y)=y;
(b) u(0,y)=1.
Solution: The characteristics will solve, with the
= 1,
x()
y()
= y,
z()
= z,
initial condition u(0, y) = f (y
1.1
WHAT IS A PARTIAL DIFFERENTIAL EQUATION?
MATH 124A Solution Key HW 01
1.1
WHAT IS A PARTIAL DIFFERENTIAL EQUATION?
2. Which of the following operators are linear? Justify your answers.
(a) L u = u x + xu y
(b) L u = u x + uu y
(c) L u = u x + u2y
(d)
Math 124A December 02, 2010
Viktor Grigoryan
18
Separation of variables: Neumann conditions
The same method of separation of variables that we discussed last time for boundary problems with Dirichlet conditions can be applied to problems with Neumann, and
Math 124A November 30, 2010
Viktor Grigoryan
17
Separation of variables: Dirichlet conditions
Earlier in the course we solved the Dirichlet problem for the wave equation on the nite interval 0 < x < l using the reection method. This required separating th
2.1
FIRST-ORDER LINEAR EQUATIONS
MATH 124A HW 02
You are encouraged to collaborate with your classmates and utilize internet resources provided you
adhere to the following rules:
(i) You read the problem carefully and make a serious effort to solve it bef
Practice Test 3
July 28, 2015
Definitions
A semi-randomly chosen 6 of the following definitions will appear on the exam.
Solution to Inhomogeneous Diffusion on the whole real line
Solution to Inhomogeneous Wave equation on the whole real line
Separated So
Simin Jin
MATH 124A
HW 2
2.1
1
3. Since we want to slove U + 1+x2 Uy = 0 which is same as (1 + x2 )Ux + Uy = 0, then we can modify it
dy
as we want to slove Ux + dx Uy = 0, then we can have the characteristic equation
(
dy
dx
du
dx
1
= 1+x2
(1)
=0
(2)
Sol
3.1
DIFFUSION ON THE HALF-LINE
Solutions prepared by Jon Tjun Seng Lo Kim Lin, TA Math 124A
MATH 124A Solution Key HW 05
3.1
DIFFUSION ON THE HALF-LINE
1. Solve u t = ku x x ; u(x, 0) = e x ; u(0, t) = 0 on the half-line 0 < x < 1.
heat
SOLUTION. By the m
1.3
FLOWS, VIBRATIONS, AND DIFFUSIONS
MATH 124A Solution Key HW 03
1.3
FLOWS, VIBRATIONS, AND DIFFUSIONS
3. On the sides of a thin rod, heat exchange takes place (obeying Newtons law of cooling flux proportional to temperature difference) with a medium of
2.1
THE WAVE EQUATION
Solutions prepared by Jon Tjun Seng Lo Kim Lin, TA Math 124A
MATH 124A Solution Key HW 04
2.1
THE WAVE EQUATION
2. Solve u t t = c 2 u x x ,
u(x, 0) = log(1 + x 2 ),
u t (x, 0) = 4 + x.
SOLUTION. An application of dAlembert formula m
3.4
WAVES WITH A SOURCE
Solutions prepared by Jon Tjun Seng Lo Kim Lin, TA Math 124A
MATH 124A Solution Key HW 06
3.4
WAVES WITH A SOURCE
2. Solve u t t = c 2 u x x + e a x ,
u(x, 0) = 0,
u t (x, 0) = 0.
SOLUTION. Let denote the characteristic triangle ab
1.1
WHAT IS A PARTIAL DIFFERENTIAL EQUATION?
MATH 124A Solution Key HW 01
1.1
WHAT IS A PARTIAL DIFFERENTIAL EQUATION?
2. Which of the following operators are linear? Justify your answers.
(a) L u = u x + xu y
(b) L u = u x + uu y
(c) L u = u x + u2y
(d)
Math 124A November 23, 2009
Viktor Grigoryan
16
Waves with a source, appendix
In the lecture we used the method of characteristics to solve the initial value problem for the inhomogeneous wave equation, utt c2 uxx = f (x, t), < x < , t > 0, u(x, 0) = (x),
Math 124A November 23, 2010
Viktor Grigoryan
16
Waves with a source
Consider the inhomogeneous wave equation utt c2 uxx = f (x, t), < x < , t > 0, u(x, 0) = (x), ut (x, 0) = (x), (1)
where f (x, t), (x) and (x) are arbitrary given functions. Similar to th
Math 124A September 28, 2009
Viktor Grigoryan
1
Introduction
Recall that an ordinary dierential equation (ODE) contains an independent variable x and a dependent variable u, which is the unknown in the equation. The dening property of an ODE is that deriv
Math 124A September 30, 2010
Viktor Grigoryan
2
First-order linear equations
Last time we saw how some simple PDEs can be reduced to ODEs, and subsequently solved using ODE methods. For example, the equation ux = 0 (1) has constant in x as its general sol
Math 124A October 05, 2010
Viktor Grigoryan
3 3.1
Method of characteristics revisited Transport equation
A particular example of a rst order constant coecient linear equation is the transport, or advection equation ut + cux = 0, which describes motions wi
Math 124A October 07, 2010
Viktor Grigoryan
4
Vibrations and heat ow
In this lecture we will derive the wave and heat equations from physical principles. These are second order constant coecient linear PDEs, which we will study in detail for the rest of t
Math 124A October 12, 2010
Viktor Grigoryan
5
Classication of second order linear PDEs
Last time we derived the wave and heat equations from physical principles. We also saw that Laplaces equation describes the steady physical state of the wave and heat c
Math 124A October 14, 2010
Viktor Grigoryan
6
Wave equation: solution
In this lecture we will solve the wave equation on the entire real line x R. This corresponds to a string of innite length. Although physically unrealistic, as we will see later, when c