Math 3150-4 PDEs for engineers (Spring 2012)

Class information

Class meets: MW 11:50am - 12:40pm
Where: MEB 2325
Textbook: N. Asmar, Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, second edition.
Prerequisites: Vector Calculus (Math 2210), ODEs and linear algebra (Math 2250 or Math 2270/2280).

Contact information

Instructor: Fernando Guevara Vasquez
Office: LCB 212
Office hours: MTW 2pm-3pm or by appointment
Phone number: +1 801-581-7467
Email: fguevara(AT)
(replace (AT) by @)


The syllabus is available here: syllabus.pdf


Weekly homeworks 30%, two midterms 20% (each), Final 30%.

Announcements [RSS]

(Thu, May 10, 2012, 04:07 PM)

  • Please check your U-mail account for the grades for the class and final.
  • Here is a histogram of the grades for the class.
    final exam (avg=130.64/150) grades (avg=B) 140 : ************* 100 : **** 130 : ********** 90 : ***************** 120 : *********** 80 : *************** 110 : ******* 70 : ****** 100 : 60 : *** 90 : ** 50 : 40 : 30 : * 20 : **** 10 : **

grade scale:
A 94, A- 88,
B 76, B+ 82, B- 70,
C 58, C+ 64, C- 52,
D 40, D+ 46, D- 34,
E 0

  • Your finals are available for pick up. You can come either tomorrow 9-2pm or some other time (provided you email me the day before so that I can expect you).

(Sat, Apr 28, 2012, 10:15 PM)

Final Exam.
  • For the final (this Wed May 2 10:30-12:30) you may bring two one-sided (or one double-sided) letter-sized handwritten cheat sheet(s).
  • Here are the solutions to Midterm 1: exam1_sol.pdf.

(Wed, Apr 25, 2012, 04:35 PM)

Homework 9 Solutions.

(Wed, Apr 25, 2012, 10:15 AM)

Practice Final Solutions.

(Fri, Apr 20, 2012, 11:03 PM)

Practice Final.

(Wed, Apr 18, 2012, 03:16 PM)

Last Week of Classes, HW9.
  • This is a friendly reminder about our final exam which is on Wed May 2 10:30-12:30 in the usual classroom. Please plan on coming 5 minutes early, the exam will start at 10:30 sharp and no extra time will be given if you arrive late.
  • Here is the schedule for the last week of classes:
    • F 04/20: Practice final and review sheet posted
    • M 04/23: Lecture on application of separation of variables to cloaking. (not included in final exam program)
    • T 04/24: Optional review/Q&A session in AEB 340 11:50-12:40. Bring questions.
    • W 04/25: Solution of practice final in class. (Last day of classes)
  • HW9: This assignment is extra credit and optional, it will be weighted as much as a regular homework. It is due in class W 04/25 and will be returned graded on the final or earlier upon request. It consists of:
    • 4.4: 10a, 13
    • 4.5: 3
  • The solutions to Midterm 2 are here: exam2_sol.pdf. The histograms for the midterm 2 grade and partial grades follows. The partial grade is computed by weighting by 30% the average of the homeworks (all counted equally and removing the lowest HW grade) and by 70% the midterm 1 and 2 average. Your actual grades will be sent to your Umail account.
    midterm 2 (avg=81.40) partial grades 2 110 : * 90 : *************** 100 : **** 80 : ************** 90 : ************* 70 : ********* 80 : ******* 60 : ****** 70 : ********** 50 : * 60 : ***** 40 : * 50 : ** 30 : **** 40 : ** 20 : * 30 : * 10 : * (updated to fix range)
  • This week we covered the Laplace equation on wedge (section 4.4) and the Laplace equation on a cylinder (section 4.5). The class notes are here: math3150_11.pdf. These topics maybe included in the final exam.

(Mon, Apr 9, 2012, 01:42 PM)

Practice Midterm Solutions.
  • Here are the solutions to the practice midterm we did in class today: exam2_practice_sol.pdf. Note that this practice exam is longer and harder than the actual exam.
  • The solutions to HW8 are here: math3150s12_hw08_sol.pdf.
  • For Wednesday’s midterm:
    • No calculators, computers or cellphones are allowed.
    • No notes or books are allowed, except for a one sided letter-sized hand-written cheat sheet.
    • Plan on arriving 5min early as the exam will start at 11:50AM and no extra time will be given if you arrive late.

(Wed, Apr 4, 2012, 11:32 PM)

Practice Midterm 2.
  • Here is the practice midterm 2: exam2_practice.pdf and the review sheet: review_sheet2.pdf (which is more specific about what is included in the exam). We will solve the practice exam together on Mon April 9.
  • There will be a Q&A/review session on Tue April 10, 11:50-12:40 in AEB 340. Attendance is not required. This session will be best used if you bring specific topics you would like to be covered.
  • This week we finished covering sections 4.3 and 4.4 in the book (the general case for vibrations of a circular membrane and Laplace’s equation in a disk). These notes are in math3150_10.pdf.
  • The solutions to HW7 are here: math3150s12_hw07_sol.pdf.
  • Here are the cores I showed this week for the wave equation on a circular membrane:
    • e4_2_2.m — Radially symmetric initial conditions
    • e4_3_2.m — General initial conditions
    • Both codes need the first few zeros of Bessel functions, which are stored in the file bz.mat (to be put in the same directory as the m-files above). If you are interested in generating these zeros I used the function besselzeros.m which can be found here (see precomp_besseljzeros.m).

(Thu, Mar 29, 2012, 09:52 PM)

Notes 10.
  • Here is the set of notes corresponding to this week: math3150_10.pdf. We finish covering the Laplace equation in rectangular domains (previous set of notes) and the wave equation in circular membranes in the case where the initial velocity is independent of angle. Some things we skipped from the notes:
    • Cylindrical and spherical coordinates (p58)
    • Discussion on Bessel function p62-64. However we did use the Bessel function identities in p63 (and back of your book) to evaluate the integrals from example 4.2.2 in p67.
    • The general case (section 4.3 p68-71) will be done next week.
  • The solutions for HW6 are here: math3150s12_hw06_sol.pdf.

(Thu, Mar 29, 2012, 09:20 AM)

Homework 8.
  • Homework 8 is due on Wed April 4 in class and consists of:
    • 3.8: 2
    • 4.1: 2,3
    • 4.2: 2,4
  • Midterm 2 will be on Wed April 11. A study sheet and a practice exam will be posted a week before the exam. The Mon April 9 lecture wil be devoted to solving the practice exam.
  • Class notes for the week will be posted soon.

(Wed, Mar 21, 2012, 03:12 PM)

Homework 7.
  • Homework 7 is due Wed March 28 in class and consists of:
    • 3.7: 2,13,16. Each problem is worth 10pt. 2b is extra credit, worth 5pts, please see the Matlab code below.
  • Here is the Matlab code we used in class:
    • An implementation of the 2D Wave Equation (example 3.7.1): e3_7_1.m
    • An implementation of the 2D Heat Equation (example 3.7.2): e3_7_2.m
  • You can have fun with the Falstad interactive applet for the 2D Wave equation which can be found here
  • The class notes for the week are here: math3150_09.pdf. We covered section 3.7 (2D wave and heat equations) and we just started with the 2D Laplace equation (section 3.8).

(Mon, Mar 12, 2012, 04:47 PM)

Other Boundary Conditions.
  • The code we saw together in class for the 1D bar with homogeneous Neumann boundary conditions (i.e. insulated at the ends) is here e3_5_1.m.
  • The code for mixed Dirichlet, Robin boundary conditions is here: e3_6_3.m. The trickery used to solve this problem is beyond the scope of this class but here is a quick description:
    • Newton’s method is applied several times with different initial conditions to find numerical approximation to the first few roots.
    • The inner products to find the generalized Fourier series coefficients of the initial condition are calculated numerically using the composite trapezoidal rule.

(Wed, Mar 7, 2012, 10:34 AM)

Homework 6.
  • Homework 6 is due on Wed 03/21 in class and consists of:
    • 3.4: 1,4
    • 3.5: 4,14,9
    • 3.6: 2
    • Extra credit: 3.4.9 (only part concerning 3.4.1)
  • Here are the class notes for the week: math3150_08.pdf. We covered the 1D Heat equation with different boundary conditions.

(Fri, Mar 2, 2012, 02:54 PM)

Notes 7.

(Wed, Feb 29, 2012, 04:35 PM)

Midterm Grades.
  • The midterm 1 was given back today. Here is a histogram for the midterm grades and a partial grade. The partial grade is computed by weighting by 30% the average of the homeworks (all counted equally and removing the lowest HW grade) and by 70% the midterm 1 grade. Your actual grades were sent to your Umail account.
    midterm 1 partial grades 1 110 : ** 100 : ****** 100 : ****** 90 : *********** 90 : ************** 80 : ********** 80 : *********** 70 : ********* 70 : ********* 60 : ******* 60 : ******* 50 : *** 50 : **** 40 : ** 30 : *

(Wed, Feb 22, 2012, 03:28 PM)

Practice Midterm Solution.
  • We will have a question/answer session on Fri Feb 24 at the usual time (11:50-12:40) in the usual classroom (MEB 2325). Please bring specific questions. It is not required that you come.
  • The Practice Midterm solution is here: exam1_practice_sol.pdf.
  • The solutions to HW5 are here: math3150s12_hw05_sol.pdf.

(Mon, Feb 20, 2012, 09:37 AM)

Practice Midterm 1.
  • The Midterm 1 is Mon Feb 27 at the usual classtime in the usual classroom.
    • Please plan on arriving 5 minutes earlier. We are starting at 11:50AM and no extra time will be given if you arrive late.
    • Calculators, cellphones or computers are not allowed (and are not needed).
    • No notes or books are allowed, except for a one-sided, letter-sized, handwritten (and not mechanically reduced) cheat-sheet.
  • The practice midterm is here: exam1_practice.pdf. We will solve this together on W Feb 22.
  • The review sheet for the midterm is here: review_sheet1.pdf.

(Fri, Feb 17, 2012, 10:33 AM)

Homework 4 Solutions.

(Wed, Feb 15, 2012, 02:38 PM)

Homework 5.
  • HW5 is due on Wed Feb 22 in class and consists of:
    • 3.3: 2, 3, 4 (Note: you don’t even need to evaluate integrals here. It is much faster to use the orthogonality conditions. See HW3 solutions below.)
  • The notes for the week are here: math3150_06.pdf. We derived the 1D wave equation and solved it using separation of variables. We will be using the same method again and again in this class, so if you understand this, you understand 90% of the class!
  • The illustration of the example we did in class (Problem 3.3.5, up to a constant) is here: p_3_3_5.m.
  • I also showed this Java applet by Falstad that simulates a vibrating string. It is really fun to play with.
  • Solutions to HW3 are here: math3150s12_hw03_sol.pdf.
  • There is no class on Mon Feb 20 (president’s day). However I will post a practice midterm exam that we will solve together in class.

(Sat, Feb 11, 2012, 11:40 PM)

Notes 5.
  • Here is what we covered last Wed: math3150_05.pdf. We looked at half range cosine and series expansion (i.e. pretending a function is even or odd).

(Wed, Feb 8, 2012, 04:18 PM)

Midterm, HW3 and HW4.
  • The due date for HW3 has been exceptionally extended to Mon 02/13.
  • HW4 is due on Wed 02/15 in class. No late homeworks accepted. It consists of:
    • 2.3: 2,7
    • 2.4: 3,6
  • The Midterm 1 will be on 02/27. Here is how we are going to use the next lectures:
    • M 02/13 – S2.4 examples + S3.2-3.3. Last day to turn in HW3.
    • W 02/15 – S3.3. HW4 due, HW3 returned, HW5 posted.
    • M 02/20 – no class (President’s day). Practice exam posted.
    • W 02/22 – solve practice exam. HW5 due, HW4 returned.
    • M 02/27 – Midterm 1: 1.1,1.2, 2.1-2.4, 3.2-3.3.
    • W 02/29 – S3.4. Midterm returned.
  • Solutions to HW2 are here: math3150s12_hw02_sol.pdf.

(Wed, Feb 8, 2012, 01:07 AM)

Notes 4.

(Wed, Feb 1, 2012, 08:29 PM)

Homework 3.
  • HW3 is due on Wed Feb 8 in class and consists of the following problems.
    • 2.1: 9
    • 2.2: 7, 9, 10a, 11a, 13
  • A couple of reminders on HW3:
    • Please include both plots and code used to generate them.
    • We will deal with what the Fourier series approximation is at discontinuity points next Monday. You may wait until the lecture to answer questions related to this (or read the class notes).
  • The solutions to HW1 are here: math3150s12_hw01_sol.pdf.

(Mon, Jan 30, 2012, 02:43 PM)

HW2 Hint.
  • I gave a hint in class about HW2 Problem 4: you need to check that the functions are 2p periodic. The inner product integrals are over the interval [-p,p] and the calculations are very similar to the calculation we did in class (which appears in the notes below).
  • The class notes for today are here: math3150_03.pdf (scan to be improved later today). They include Fourier series and some mathematical definitions we need to continue. Note that we started on p17.

(Thu, Jan 26, 2012, 10:48 AM)

Lecture Notes 2.

The lecture notes for the week are here: math3150_02.pdf. We talked about inner products in R^n, this is essentially a review of concepts from Vector Calculus, but that will be very useful to us when doing Fourier series approximations.

(Wed, Jan 25, 2012, 05:16 PM)

Homework 2.
  • Homework 2 is here hw2.pdf and is due on Wed Feb 1st in class. Please note that the case of p=pi will be done in class on Monday Jan 30.
  • The code from the computer lab is here:
  • Class notes for the week will be posted soon.

(Mon, Jan 23, 2012, 09:28 AM)

Computer Lab.
  • Today we meet in LCB 115 for a Computer Lab. The program is here: cl01.pdf. The purpose of this computer lab is for you to be familiar with Matlab, especially to plot functions. The code we generated will be posted in the class website later today.
  • The Math department provides free tutoring for this class in the T. Benny Rushing Mathematics Center. The tutors that are most likely to help you with Math 3150 are: Brandon S, Caleb, Geoff, Justin, Matt, Trevor D. For their schedules and other details, please visit the Mathematics Tutoring Center.
  • To use Matlab or Octave you may:

(Wed, Jan 18, 2012, 09:24 AM)

Homework 1.
  • HW1 is due on Wed Jan 25 in class. No late homeworks will be accepted. The homework consists of the following:
    • 1.1: 11
    • 1.2: 3, 5a, 7, 16

(Thu, Jan 12, 2012, 02:36 PM)

Lecture Notes.
  • Here are the lecture notes for the week: math3150_01.pdf.
  • The first homework will be posted on or before Jan 18.
  • We have a computer lab on Mon Jan 23 in LCB 115 during class time.
    • We will get acquainted with the computing facilities so that everyone has access to Matlab. A login has been created for you at the Math department system. If you haven’t done so, here are the instructions on how to login.
    • There may not be enough workstations for all of us. So if you are familiar with Matlab or if you will be using another software for this class it is not necessary to come. You may also bring your laptop if you prefer.
    • Using Matlab is not required for this class. You may use other software for the occasional plots in your assignments, however all sample code for this class will be given in Matlab. You may obtain a student license (for about ~$100) at the University Bookstore or you may use the free alternative Octave. Here are more detailed installation instructions.
  • Note: These announcements are sent to your UMail ( account. If you prefer to receive them at another email address, please forward your UMail by following the instructions at the UMail webpage.