UBC Math 257: Partial Differential Equations
This page provides official background information
for two courses running in parallel:
- UBC Math 257 — Partial Differential Equations, and
- UBC Math 316 — Elementary Differential Equations II.
Daily readings, homework, and updates, are on the
courses' Unified Home Page.
Dr. Philip D. Loewen
||Mathematics Building, room 207
(try this first)
||604-822-3082 (urgent cases only, please)
||I'm happy to meet individuals or small groups in room MATH 207,
but advance planning is essential. To schedule a conversation, email
including your course number (e.g., “264” or “211”)
in the subject line.
|Review of Series
- Convergence concepts
- Geometric series and the ratio test
- Error control and Shanks Transformation
- Power series
- Analytic functions, radius of convergence
|Linear Ordinary Differential Equations
- Power Series Solutions near an ordinary point
- General Solutions
- Initial-Value Problems
- Non-homogeneous problems
- Homogeneous ODE's with Constant Coefficients
- Non-Homogeneous ODE with Constant Coefficients
- ODE's of Euler type
- Change of independent variable
- Singular points: regular vs irregular
- Series solutions near a regular singular point (Frobenius method)
|Linear Operators and Eigenvalue Problems
- The matrix case
- Operators and eigenvalues in function space
- Sample eigenvalue problems
- Orthogonality and eigenfunction series
|Eigenfunction-series solutions for Boundary-Value Problems
- Separation of variables: eigenvalue problems, formal series
- The Big Four standard eigenvalue problems
- Propagating coefficients
- Special modes; superposition
- Nonhomogeneous PDE
- Nonhomogeneous BC
|More General Eigenfunction Series
- Eigenvalue Problems
- Orthogonality and Eigenfunction Series
- Sturm-Liouville Theory
- Full Fourier Series
|Heat Equation in Depth
- Derivation; BC's and their meanings
- Transient versus long-run behaviour
- Splitting methods for nonhomogeneous boundary conditions
|Laplace's Equation in Depth
- Derivation and Interpretation
- Superposition and splitting
- Neumann problems and consistency conditions
- Physical solutions are bounded functions
- Problems in polar coordinates (Euler equations reappear)
|Wave Equation in Depth
- Derivation; BC's and their meanings
- Travelling wave solutions
- D'Alambert's solution
- Signal speed; reflections
- Forcing and resonance
- Finite Differences
- Spreadsheet Implementations
|Total Hours on Outline
- (44 lecture-hours + 2 test-hours = 46 meetings this term)
- William F. Trench, Elementary
Differential Equations with Boundary Value Problems, 2013.
Book 9 in the collection Books
and Monographs available free online at
Watch the main course page for writeups customized for
this term's experience. In addition, consult the following:
- W. E. Boyce and R. C. DiPrima,
Elementary Differential Equations and Boundary Value Problems,
any edition since about #5. New York: John Wiley & Sons, 1997.
- Froese, Richard G.,
Partial Differential Equations.
UBC M257/316 Lecture notes free on the Web.
- Peirce, Anthony,
course page for Sep-Dec 2013.
Look especially at the section headed “Lecture Notes”.
Other Nice Books:
- Churchill, R. V., and J. W. Brown, Fourier Series and Boundary Value
Problems. New York: McGraw-Hill, 1993.
- Troutman, John L., Boundary Value Problems of Applied Mathematics.
Boston: PWS Publishing Company, 1994.
- Main, Iain G., Vibrations and Waves in Physics, third edition.
Cambridge University Press, 1993.
- Tuesday 13 May: Class starts, 14:00 in room LSK 200.
- Thursday 19 June: Midterm test in class. 110 minutes.
- Friday 20 June – Wednesday 2 July: no classes.
- Thursday 3 July: Classes resume after mid-session break.
- Thursday 7 August: Last class of the term.
- Wednesday 13 August: Final exam, 19:00–21:30, room TBA.
(This info is unofficial. Please confirm it on
Enrolment Services Exam Schedule.)
- Daily Participation (bring your iClicker to class), 0-2%.
- Weekly Homework (Due every Tuesday), 10%.
- 19 June 2014 (Thursday), Midterm Exam, 38-40%, in class.
- 12–16 August 2014, Final Exam, 50%, scheduled by UBC Enrolment Services.
Students may use
no resources except for writing equipment
on midterm and final examinations. This means
no formula sheet and
no calculator. Seriously.
However, students are encouraged to use technology to help with weekly
The final course grade is influenced only by what knowledge of
the subject students demonstrate in the activities described
There is no supplemental examination in this course.
Missing a midterm normally results in a mark of 0. Exceptions may be
granted in two cases: prior consent of the instructor or a medical
emergency. In the latter case, the instructor must be notified within 48
hours of the missed test, and presented with a doctor's note immediately
upon the student's return to UBC.
On both homework and tests, the highest possible standard of
academic integrity is expected and will be enforced. In short,
students are expected to submit work that represents their own
current thoughts and understanding. For details, review the
UBC Calendar's sections on
Academic Honesty and Standards
Course Web Page (daily details appear here):
Last update: 07 Aug 2014 (Thu), 11:47:48.
(Click a graphic to recheck.)