The Pierce Lab

ACM 95b/ACM 100b

Introductory Methods of Applied Mathematics

Instructor

Niles A. Pierce
165 Broad
Open office hours

Overview

The second term focuses on solution methods for linear ordinary differential equations.

Topics

IVPs

Introduction to differential equations, linear first order ODEs, integrating factors, integral curves, singular points, existence and uniqueness, the view in the complex plane, second order linear initial value problems (IVPs), solution properties for homogeneous problems, solutions in the constant coefficient case, reduction of order, nonhomogeneous initial value problems, variation of parameters, the delta function, Heaviside step function, Green's functions for initial value problems, jump conditions, the Laplace transform, convolution, shifting theorems, inverting Laplace transforms with the Mellin inversion formula and the Bromwich contour, nonlinear first order ODEs, Picard's existence and uniqueness theorem, numerical methods, explicit Euler, implicit Euler and trapezoidal rule, truncation error, order of accuracy, solution error, explicit Runge-Kutta methods, linear stability analysis, generalization to first order nonlinear systems, linear equations with analytic coefficients, series solutions near ordinary points, solution behavior near singular points, regular singular points, Euler equations, solutions near regular singular points by the method of Frobenius.

BVPs

Second order linear boundary value problems (BVPs), eigenvalue problems, eigenfunctions, Fourier series, Fourier cosine series, Fourier sine series, solution of the heat equation by separation of variables and Fourier series, differentiation of Fourier series, complex Fourier series, the Fourier transform, Parseval's theorem, convolution, solution of the heat equation on an infinite domain, the adjoint problem, self-adjointness, regular Sturm-Liouville eigenvalue problems, properties of eigenvalues and eigenfunctions, the Rayleigh quotient, eigenfunction expansions, nonhomogenous boundary value problems, the Fredholm alternative, Green's functions for boundary value problems, jump conditions, reciprocity, singular Sturm-Liouville eigenvalue problems, Bessel functions, solution of the radially-symmetric heat equation in a circular domain.

Reserve Texts

  • G. Birkhoff and G.-C. Rota. Ordinary Differential Equations, 3rd ed, Wiley, 1978.
  • W.E. Boyce and R.C. DiPrima. Elementary Differential Equations and Boundary Value Problems, 7th ed, Wiley, 2001.
  • E. Butkov, Mathematical Physics, Addison-Wesley, 1968.
  • E.A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall, 1961.
  • R. Haberman, Elementary Applied Partial Differential Equations, 3rd ed, Prentice Hall, 1998.
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, 1991.
  • H.F. Weinberger, A First Course in Partial Differential Equations with Complex Variables and Transform Methods, Dover, 1965.

Problem Sets

  • Available online at 3pm
  • Due at 3pm in the slot of Firestone 303
  • Collaboration is encouraged but prepare your own unique solutions
  • If you need help, ask a TA...
  • If you still need help, ask the instructor
  • Extensions only in exceptional circumstances: see the Head TA
  • Accepted without extension for 50% credit up to one week late
  • Solutions available online immediately after the due date
  • Please report suspected errors in problems or solutions to the Head TA

Exams

  • Available online at 3pm
  • Due at 3pm in the slot of Firestone 303
  • Exams should be completed in a single sitting of 4 hours
  • Calculators are not permitted
  • Closed-book: with the exception of official lecture handouts and problem set questions, only material written in your own hand or typed by your own hand may be used during exams
  • Extensions only with permission of the Dean
  • A review session will be offered before each exam

Grading

25% problem sets, 37.5% midterm, 37.5% final

Problem Set & Exam Schedule

Available Due
PS 1Monday 1/8Wednesday 1/17
PS 2Wednesday 1/17Wednesday 1/24
PS 3Wednesday 1/24Friday 2/2
MidtermFriday 2/2Wednesday 2/7
PS 4Friday 2/9Friday 2/16
PS 5Friday 2/16Friday 2/23
PS 6Friday 2/23Friday 3/2
PS 7Friday 3/2Friday 3/9
FinalWednesday 3/14Friday 3/16

Tutorial Sections

Attend any section of your choosing. Please leave your official section unchanged.

Section Time Location
A2pm Mon119 Downs
B11am Tues119 Downs
C1pm Tues11 Downs
D2pm Wed119 Downs
E11am Thurs119 Downs
F2pm Thurs11 Downs
G1pm Fri103 Downs

Teaching Assistants

Head TA: Lei Zhang and the ACM95/100 Underground

Name Office hour Location Section
A
B
C
D
E
F
G