welcome to math 257/316 - partial differential...
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Welcome to Math 257/316 - Partial Differential Equations
• Instructor: Mona Rahmani • email: [email protected] • Office: Mathematics Building 110 • Office hours: Mondays 2 - 3 pm, Wednesdays and
Fridays 1 - 2 pm. • Course webpage: https://blogs.ubc.ca/mrahmani/
teaching/math257-316-2018/ (all assignments and lecture notes will be posted here.)
• A list of course topics is given in the course outline on the webpage.
List of Topics1. Review of techniques to solve ODEs 2. Series solutions of variable coefficient ODEs 3. Introduction to partial differential equations
a. The heat equationb. The wave equationc. Laplace’s equation
4. Introduction to numerical methods for PDEs 5. Fourier series and separation of variables. a. The heat equation b. The wave equation
c. Laplace’s equation6. Boundary value problems and Sturm-Liouville theory
Text book (recommended but not required):• Elementary Differential Equations and Boundary Value Problems by
Boyce & DiPrima, Etext.• Applied Partial Differential Equations with Fourier Series and Boundary
Value Problems (4nd Ed), R. Haberman, (Pearson), 2004. •• Partial Differential Equations for Scientists and Engineers (1st Ed), S.
Farlow, (Dover), 1993.
Online resources:
• Professor Anthony Peirce’s course material: https://www.math.ubc.ca/~peirce/ you can find all last terms lecture notes, previous assignments, exams and their solutions on this webpage.
• Professor Richard Froese’s lecture notes: http://www.math.ubc.ca/~rfroese/notes/Lecs316.pdf
Formula sheet - also posted on the course webpage:
Math 257-316 PDE Formula sheet - final exam
Trigonometric and Hyperbolic Function identities
sin(↵ ± �) = sin ↵ cos � ± sin� cos ↵ sin2 t + cos
2 t = 1
cos(↵ ± �) = cos ↵ cos � ⌥ sin� sin↵. sin2 t =
12 (1� cos(2t))
sinh(↵ ± �) = sinh↵ cosh� ± sinh� cosh↵ cosh2 t� sinh
2 t = 1
cosh(↵ ± �) = cosh↵ cosh� ± sinh� sinh↵. sinh2 t =
12 (cosh(2t)� 1)
Basic linear ODE’s with real coe�cients
constant coe�cients Euler eqODE ay00 + by0 + cy = 0 ax2y00 + bxy0 + cy = 0
indicial eq. ar2+ br + c = 0 ar(r � 1) + br + c = 0
r1 6= r2 real y = Aer1x+ Ber2x y = Axr1 + Bxr2
r1 = r2 = r y = Aerx+ Bxerx y = Axr
+ Bxrln |x|
r = � ± iµ e�x[A cos(µx) + B sin(µx)] x�
[A cos(µ ln |x|) + B sin(µ ln |x|)]
Series solutions for y00 + p(x)y0 + q(x)y = 0 (?) around x = x0.
Ordinary point x0: Two linearly independent solutions of the form:
y(x) =P1
n=0 an(x� x0)n
Regular singular point x0: Rearrange (?) as:
(x� x0)2y00 + [(x� x0)p(x)](x� x0)y0 + [(x� x0)
2q(x)]y = 0
If r1 > r2 are roots of the indicial equation: r(r � 1) + br + c = 0 where
b = limx!x0
(x� x0)p(x) and c = limx!x0
(x� x0)2q(x) then a solution of (?) is
y1(x) =P1
n=0 an(x� x0)n+r1 where a0 = 1.
The second linerly independent solution y2 is of the form:
Case 1: If r1 � r2 is neither 0 nor a positive integer:
y2(x) =
1X
n=0
bn(x� x0)n+r2 where b0 = 1.
Case 2: If r1 � r2 = 0:
y2(x) = y1(x) ln(x� x0) +
1X
n=1
bn(x� x0)n+r2 for some b1, b2...
Case 3: If r1 � r2 is a positive integer:
y2(x) = ay1(x) ln(x� x0) +
1X
n=0
bn(x� x0)n+r2 where b0 = 1.
Fourier, sine and cosine series
Let f(x) be defined in [�L,L]then its Fourier series Ff(x) is a 2L-periodic
function on R: Ff(x) =a02 +
P1n=1
�an cos(
n⇡xL ) + bn sin(
n⇡xL )
where an =1L
R L�L f(x) cos(
n⇡xL ) dx and bn =
1L
R L�L f(x) sin(
n⇡xL ) dx
Theorem (Pointwise convergence) If f(x) and f 0(x) are piecewise con-
tinuous, then Ff(x) converges for every x to12 [f(x�) + f(x+)].
Parseval’s indentity
1
L
Z L
�L|f(x)|2dx =
|a0|2
2+
1X
n=1
�|an|2 + |bn|2
�.
For f(x) defined in [0, L], its cosine and sine series are
Cf(x) =a0
2+
1X
n=1
an cos(n⇡x
L), an =
2
L
Z L
0f(x) cos(
n⇡x
L) dx,
Sf(x) =
1X
n=1
bn sin(n⇡x
L), bn =
2
L
Z L
0f(x) sin(
n⇡x
L) dx.
D’Alembert’s solution to the wave equation
PDE: utt = c2uxx, �1 < x <1, t > 0 IC: u(x, 0) = f(x), ut(x, 0) = g(x).
SOLUTION: u(x, t) =12 [f(x + ct) + f(x� ct)] +
12c
R x+ctx�ct g(s)ds
Sturm-Liouville Eigenvalue Problems
ODE: [p(x)y0]0 � q(x)y + �r(x)y = 0, a < x < b.BC: ↵1y(a) + ↵2y0(a) = 0, �1y(b) + �2y0(b) = 0.
Hypothesis: p, p0, q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for
x 2 [a, b]. ↵21 + ↵2
2 > 0. �21 + �2
2 > 0.
Properties (1) The di↵erential operator Ly = [p(x)y0]0� q(x)y is symmetric
in the sense that (f, Lg) = (Lf, g) for all f, g satisfying the BC, where (f, g) =R ba f(x)g(x) dx. (2) All eigenvalues are real and can be ordered as �1 < �2 <
· · · < �n < · · · with �n !1 as n!1, and each eigenvalue admits a unique
(up to a scalar factor) eigenfunction �n.
(3) Orthogonality: (�m, r�n) =R b
a �m(x)�n(x)r(x) dx = 0 if �m 6= �n.
(4) Expansion: If f(x) : [a, b]! R is square integrable, then
f(x) =
1X
n=1
cn�n(x), a < x < b , cn =
R ba f(x)�n(x)r(x) dxR b
a �2n(x)r(x) dx
, n = 1, 2, . . .
1
You will get this formula sheet on the exams. Please make sure you become familiar with the notation as we progress through the term.
Grades:
• Final exam: 50%. Students must get at least 35% on the final exam to pass the course.
• Two in-class midterm exams (each 20%): 40%. There will be no make-up midterms. If you cannot make it to any of the midterms (for a legitimate reason), you must inform me at least two days before the test date.
• Homework (including Matlab assignments): 10%. Assignment should be submitted at the beginning of the class on the day they are due. No late submission or electronic submission will be accepted. You must submit your assignment at the section you are registered in. The handed in assignment must be your own work.
Midterm dates (two in-class midterms):
Friday, February 15
Wednesday, March 20
Matlab:
• Some homework include Matlab assignments. The scripts will be provided. However, the assignments may require some modification of the scripts.
• Matlab is free for all UBC students. See UBC IT services.
