Partial Differential Equations for Engineers - Assignment 2 | MATH 3150, Assignments of Mathematics

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Partial Differential Equations for Engineers

Math 3150

University of Utah

Spring 2002

Benjamin McKay

[email protected]

Office: JWB 126

Telephone: 581-

April 29, 2002

Contents

11.7 Half lines........................... 146 11.7.1 Derivatives..................... 147 11.7.2 Heat in a half infinite wire............ 148

III

1 About this document

This document resides in

http://www.math.utah.edu/∼mckay/3150.html

(the course web page). It provides the course notes for Math 3150: PDEs for Engineers, and will be developed as the course continues, providing some notes on the lectures; but the homework assignments given below will probably not change—if they do, you will be notified in class.

2 Structure of the course

2.1 Students with disabilities

The University of Utah seeks to provide equal access to its programs, services and activities for people with disabilities. If you will need accomodations in this class, reasonable prior notice needs to be given to the instructor and to the Center for Disability Services, 162 Olpin Union Building, 581-5020 (V/TDD) to make arrangements for accomodations. All written information in this course can be made available in alternative format with prior notification.

2.2 What you know already

• 3 semesters of calculus, including vector or multivariable calculus
• Ordinary differential equations and linear algebra (as in the 2250 course, a 3 credit course with only a little linear algebra, mostly matrices, hopefully a touch of abstract vector spaces)

2.3 Core material

• Wave, heat and Laplace (=electrostatics) equations
• Changing coordinates and separating variables
• Fourier series and transforms

2.4 The textbook

The book is

Naklh´e Asmar, Partial Differential Equations and Boundary Value Problems, Prentice–Hall, Upper Saddle River, NJ, 2000.

which is available from the book store for $82 new and $65 used. Previously, this course used

2.6.1 Some help with Maple

1. There are computer labs in South Physics 205 and in EMCB. For more information, see

www.math.utah.edu/ugrad/lab/.

1. To start Maple, type xmaple

2. To get help in Maple, look at the Help menu. Maple’s help facility gives examples. Cut and paste to try them out.

3. Maple commands must end with a semicolon

;

1. To get Maple to execute a command, you have to press return. Go to any line of the worksheet, and press return there, to have that line executed again.

2. Comments start with

1. Maple distinguishes uppercase and lowercase letters; for example π is writ- ten as Pi, not pi or PI.

2. Watch out for multiplication signs, written

You need to write them all the time. For instance,

2x

means nothing; you must type

2*x

1. When Maple does something that doesn’t make sense to you, try the Edit menu, under Execute, and select Worksheet. This will get Maple to start all over from the beginning. You might also find that it helps to put a

restart;

statement wherever you want to clear out all of the old variables, and at the beginning of the file.

1. If you have already used a variable for something else (like m in our case, which gets used quite a bit in bm amplitudes), you need to wipe out its old values before using it again. For example, to wipe out the previous values of variables m and x type

m := ’m’; x := ’x’;

1. You can save your work into a file in several formats. Generally, use the default format (Maple Worksheet). Look in the File menu under Save As (or Save if you have previously saved the file).

2. To print, use the File menu Print command.

3. Maple has a tutorial, in the Help menu under New User’s Tour.

4. There is a short tutorial for Maple in

www.math.utah.edu/∼korevaar

under the Math 2250 selection.

2.7 Tutoring

Free tutoring is available in Mines 210 (Mines is north of INSCC), available everyday, except weekends and holidays. Hours are posted on

http://www.math.utah.edu/ugrad/tutoring.html

Tutoring is also available through the University of Utah Tutoring Center, in the Student Services Building, room 330. Cost is $6.00 per hour. Students are given a list of tutors to contact and schedule a day, evening or weekend appointment. Low income students may qualify for free tutoring. For more information, call 581-5153 or visit www.saff.utah.edu/Tutoring/.

2.8 What we cover, and when and where

2.8.1 Lectures: where and when

Section 1 Mondays and Wednesdays 10:45am–11:35am OSH 107 Section 2 Tuesdays and Thursdays 9:40am–10:30am BU C 301

2.8.2 Office hours: where and when

I will be in my office, which is JWB 126 (in the basement of the John Widtsoe Building, on President’s circle) 8:30am–9:30am, Monday–Thursday, and you can drop by then to ask questions; or you can schedule an appointment with me.

3

Monday/Wednesday (Section 1) Schedule

Section 1 Schedule

Date

Topic

Textbook

Assignment

Mon, Jan 7

Why learn PDE?

Wed, Jan 9

Periodicity & Fourier series

Mon, Jan 14

Playing with Fourier series

HW #1 & #2 due

Wed, Jan 16

Energy & Parseval’s iden-tity

HW #3 due

Mon, Jan 21

Martin Luther KingDay (no classes)

Wed, Jan 23

Complex Fourier series

HW #4 due

Mon, Jan 28

Oscillators

HW #5 due

Wed, Jan 30

Test #

Wed, Feb 27

Waves & strings

HW #6 due

Mon, Mar 4

Separating variables

HW #7 due

Wed, Mar 6

d’Alembert’s method

HW #8 due

Mon, Mar 11

Heat

HW #9 due

Wed, Mar 13

Hot bars

HW #10 due

Mon, Mar 18

Heat & waves in squareplates

HW #11 due

Wed, Mar 20

Test #

Mon, Mar 25

Changing coordinates

HW

Wed, Mar 27

Waves & heat in disks

HW #13 due

Mon, Apr 1

Steady states of cylinders& disks

HW

Wed, Apr 3

Bessel functions

HW #15 due

Section 1 Schedule (continued)

Date

Topic

Textbook

Assignment

Mon, Apr 8

Hanging chains

HW #16 due

Wed, Apr 10

Buckling beams

HW #17 due

Mon, Apr 15

More buckling beams

HW #18 due

Wed, Apr 17

Test #

Mon, Apr 22

Fourier transforms

HW #19 due

Wed, Apr 24

Heat & waves in infinitespace

HW #20 due

Mon, Apr 29

Convolution

HW #21 due

Wed, May 1

The heat kernel

HW #22 due

Section 2 Schedule (continued)

Date

Topic

Textbook

Assignment

Tues, Apr 9

More buckling beams

HW #18 due

Thurs, Apr 11

Test #

Tues, Apr 16

Fourier transforms

HW #19 due

Thurs, Apr 18

Heat & waves in emptyspace

HW #20 due

Tues, Apr 23

Convolution

HW #21 due

Thurs, Apr 25

The heat kernel

HW #22 due

Tues, Apr 30

The Poisson integralformula

HW #23 due

Thurs, May 2

Cosine & sine transforms,half lines

HW #24 due

5 Homework Assignments

Homework #1 Why learn PDE? Due on Jan 14 for Monday/Wednesday section.

Due on Jan 8 for Tuesday/Thursday section.

1. Which functions satisfy which equations? (Show your calculations.)

(a) u = sin(x − ct) (1)

∂u ∂t

= c^2

∂^2 u ∂x^2

(heat)

(b) u = e−c

(^2) t sin(x) (2) ∂^2 u ∂t^2

= c^2 ∂^2 u ∂x^2

(wave)

(c) u = x − t (3)

∂^2 u ∂x^2

= 0 (Laplace)

1. Show that the solutions u(x, y) to the PDE

y

∂u ∂x

− x

∂u ∂y

are constant along all circles around the origin of coordinates. Hint: to move along a circle of radius r, take

x = r cos θ y = r sin θ

and think of r as a constant, and θ as the variable. You will also need the chain rule for functions of several variables: d dt

f (x(t), y(t)) =

∂f ∂x

dx dt

∂f ∂y

dy dt

1. The definition of derivative is dy dx

y(x + ∆x) − y(x) ∆x for small ∆x. (The ∼ symbol means “is close to”.) Use this to show that

d^2 y dx^2

y(x + ∆x) − 2 y(x) + y(x − ∆x) (∆x)^2

1. The heat equation ∂u ∂t

= c^2

∂^2 u ∂x^2

0

1

20 40 60 80 100

(a) Initial temperature profile

0

1

2

20 40 60 80 100

(b) After one time step

0

200000

400000

20 40 60 80 100

(c) After five time steps

Figure 1: A naive numerical approach to the heat equation

0

1

0.2 0.4 0.6 0.8 1 x

Figure 2: The exact solution of the heat equation

Homework #2 Periodicity and Fourier series Due on Jan 14 for Monday/Wednesday section.

Due on Jan 10 for Tuesday/Thursday section.

1. Calculate (^) ∫ 2 π+ √ 2+ √ 17 √ 2+ √ 17

(1 + sin x) dx

precisely (no decimal approximations, and show how to do it—don’t just write the answer).

1. Textbook page 20, #2,3,15, page 32 # 5–9: derive the Fourier series rep- resentations by hand, not by computer. Get Maple to graph the functions, and plot the partial sums.

Homework #3 Playing with Fourier series Due on Jan 16 for Monday/Wednesday section.

Due on Jan 15 for Tuesday/Thursday section.

1. Textbook, page 41, #1–5: ignore the “points of discontinuity”—just derive the Fourier series.

2. Textbook, page 48, #1–5: plot the partial sums of 1,5 and 10 terms; you don’t have to comment on the graphs.

Homework #4 Energy and Parseval’s theorem Due on Jan 23 for Monday/Wednesday section.

Due on Jan 17 for Tuesday/Thursday section.

1. Textbook, section 3.4 #1,2,8.

2. Write code in Maple to take functions f (x) (initial position) and g(x) (initial velocity) and a length L and

(a) plot f (x) (b) plot an animation of the solution u(x, t) to the wave equation as on page 104 of the textbook. (c) Try out your animation on the initial conditions given in each of the problems 1,2,8 from section 3.4. (d) Print out the Maple worksheet showing the final state of the string at the end of each of these animations.

Homework #10 Heat Due on Mar 13 for Monday/Wednesday section.

Due on Mar 7 for Tuesday/Thursday section.

1. Textbook 3.5 #1,3,6,7.

Homework #11 Hot bars Due on Mar 18 for Monday/Wednesday section.

Due on Mar 12 for Tuesday/Thursday section.

1. Textbook 3.6 #1,3,7,16.

Homework #12 Heat & waves in square plates Due on Mar 25 for Monday/Wednesday section.

Due on Mar 19 for Tuesday/Thursday section.

1. Textbook 3.7 #1,6.

2. Write Maple code to produce a 3D animation of the solution to textbook problem 3.7 #6.

Homework #13 Changing coordinates Due on Mar 27 for Monday/Wednesday section.

Due on Mar 21 for Tuesday/Thursday section.

1. Textbook 3.8 #1,2.

2. Textbook 4.1 #1,3,9.

Homework #14 Waves & heat in disks Due on Apr 1 for Monday/Wednesday section.

Due on Mar 26 for Tuesday/Thursday section.

1. Textbook 4.2 #1,3,6.

1. Textbook 4.3 #1,2,3.

2. Write Maple code to animate the solution to textbook problem 4.3 #3.

Homework #15 Steady states of cylinders & disks Due on Apr 3 for Monday/Wednesday section.

Due on Mar 28 for Tuesday/Thursday section.

1. Textbook 4.4 #2,3,7.

2. Textbook 4.5 #1,2,5.

Homework #16 Bessel functions Due on Apr 8 for Monday/Wednesday section.

Due on Apr 2 for Tuesday/Thursday section.

1. Textbook 4.7 #3,6,

2. Textbook 4.8 #3,9,23,31.

Homework #17 Hanging chains Due on Apr 10 for Monday/Wednesday section.

Due on Apr 4 for Tuesday/Thursday section.

1. Textbook 6.3 #1,4. Also get Maple to animate the chain for each of these examples.

Homework #18 Buckling beams Due on Apr 15 for Monday/Wednesday section.

Due on Apr 9 for Tuesday/Thursday section.

1. Textbook 6.5 #2.

2. In Maple, create an animation of the vibrating beam described in textbook problem 6.5 #2.

Homework #19 More buckling beams Due on Apr 22 for Monday/Wednesday section.

Due on Apr 16 for Tuesday/Thursday section.

1. Textbook 6.5 #4,6.

2. In Maple, create an animation of the vibrating beam described in textbook problems 6.5 #4,6.

Homework #20 Fourier transforms Due on Apr 24 for Monday/Wednesday section.

Due on Apr 18 for Tuesday/Thursday section.