Math 257/316 Section 202
Name:
Midterm 2
Student #:
March 13
2 questions;
50 minutes;
max = 30 points;
Instructions: no calculators, books, notes, or electronics. Show and explain all your work.
1. (15 points) Find the solution of the non-homogeneous initial-boundary-value problem
8
>
< ut = uxx + sin(x), 0 < x < ⇡, t > 0,
u(0, t) = 0, u(⇡, t) = 1
>
:
u(x, 0) = 0
1
(Blank page)
2
2. (15 points) For the following inhomogeneous heat equation problem with mixed BCs:
8
t
>
< ut = uxx xe , 0 < x < 1, t > 0
u(0, t) = 1, ux (1, t) = 1
>
:
u(x, 0) = 0
(a) (7 points) What are the “eigenfunctions” for this problem? Write the form
of the solution u(x, t) using an eigenfunction expansion (but do not try to
compute any coefficients).
(b) (8 points) Write a finite-di↵erence numerical scheme to approximate the solution
u(x, t), using step sizes x (in space) and t (in time), with xn = n x, tk =
k t, n = 0, 1, 2, . . . , N ; k = 0, 1, 2, . . .). Include formulas for computing the
approximate values ukn ⇡ u(xn , tk ), and for incorporating the BCs and IC.
3
(Blank page)
4
Formula sheet - final exam
constant coefficients
ay 00 + by 0 + cy = 0
ar2 + br + c = 0
y = Aer1 x + Ber2 x
y = Aerx + Bxerx
e x [A cos(µx) + B sin(µx)]
Euler eq
ax2 y 00 + bxy 0 + cy = 0
ar(r 1) + br + c = 0
y = Axr1 + Bxr2
y = Axr + Bxr ln |x|
x [A cos(µ ln |x|) + B sin(µ ln |x|)]
sin2 t + cos2 t = 1
sin t = 12 (1 cos(2t))
cosh2 t sinh2 t = 1
2
sinh t = 12 (cosh(2t) 1)
2
y1 (x) =
n=0
P1
an (x
x!x0
x0 )
n+r1
where a0 = 1.
r2 = 0:
bn (x
n=1
1
X
x0 ) +
n=0
1
X
bn (x
bn (x
x0 )n+r2 where b0 = 1.
x0 )n+r2 for some b1 , b2...
x0 )n+r2 where b0 = 1.
r2 is a positive integer:
x0 ) +
n=0
1
X
y2 (x) = ay1 (x) ln(x
Case 3: If r1
y2 (x) = y1 (x) ln(x
Case 2: If r1
y2 (x) =
The second linerly independent solution y2 is of the form:
Case 1: If r1 r2 is neither 0 nor a positive integer:
x!x0
Regular singular point x0 : Rearrange (?) as:
(x x0 )2 y 00 + [(x x0 )p(x)](x x0 )y 0 + [(x x0 )2 q(x)]y = 0
If r1 > r2 are roots of the indicial equation:
r(r 1) + br + c = 0 where
b = lim (x x0 )p(x) and c = lim (x x0 )2 q(x) then a solution of (?) is
Ordinary point x0 : Two linearly independent solutions of the form:
P1
y(x) = n=0 an (x x0 )n
Series solutions for y 00 + p(x)y 0 + q(x)y = 0 (?) around x = x0 .
ODE
indicial eq.
r1 6= r2 real
r1 = r 2 = r
r = ± iµ
Basic linear ODE’s with real coefficients
sin(↵ ± ) = sin ↵ cos ± sin cos ↵
cos(↵ ± ) = cos ↵ cos ⌥ sin sin ↵.
sinh(↵ ± ) = sinh ↵ cosh ± sinh cosh ↵
cosh(↵ ± ) = cosh ↵ cosh ± sinh sinh ↵.
Trigonometric and Hyperbolic Function identities
Math 257-316 PDE
1
n=1
1
X
bn sin(
n⇡x
),
L
bn =
2
L
0
Z
L
f (x) sin(
ODE: [p(x)y 0 ]0 q(x)y + r(x)y = 0, a < x < b.
0
BC:
↵1 y(a) + ↵2 y 0 (a) = 0,
1 y(b) + 2 y (b) = 0.
0
Hypothesis: p, p , q, r continuous on [a, b]. p(x) > 0 and r(x) > 0 for
x 2 [a, b]. ↵12 + ↵22 > 0. 12 + 22 > 0.
Properties (1) The di↵erential operator Ly = [p(x)y 0 ]0 q(x)y is symmetric
in the sense that (f, Lg) = (Lf, g) for all f, g satisfying the BC, where (f, g) =
Rb
f (x)g(x) dx. (2) All eigenvalues are real and can be ordered as 1 < 2 <
a
· · · < n < · · · with n ! 1 as n ! 1, and each eigenvalue admits a unique
(up to a scalar factor) eigenfunction n .
Rb
(3) Orthogonality: ( m , r n ) = 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
Rb
1
X
f (x) n (x)r(x) dx
f (x) =
cn n (x), a < x < b , cn = a R b
, n = 1, 2, . . .
2 (x)r(x) dx
n=1
a n
Sturm-Liouville Eigenvalue Problems
n⇡x
) dx.
L
PDE: utt = c2 uxx , 1 < x < 1, t > 0 IC: u(x, 0) = f (x), ut (x, 0) = g(x).
R x+ct
1
SOLUTION: u(x, t) = 12 [f (x + ct) + f (x ct)] + 2c
g(s)ds
x ct
D’Alembert’s solution to the wave equation
Sf (x) =
For f (x) defined in [0, L], its cosine and sine series are
Z
1
a0 X
n⇡x
2 L
n⇡x
Cf (x) =
+
an cos(
), an =
f (x) cos(
) dx,
2
L
L
L
0
n=1
Theorem (Pointwise convergence) If f (x) and f 0 (x) are piecewise continuous, then F f (x) converges for every x to 12 [f (x ) + f (x+)].
Parseval’s indentity
Z
1
1 L
|a0 |2 X
|f (x)|2 dx =
+
|an |2 + |bn |2 .
L L
2
n=1
Let f (x) be defined in [ L, L]then
a 2L-periodic
P1its Fourier series F f (x) isn⇡x
function on R: F f (x) = a20 + n=1 an cos( n⇡x
)
+
b
sin(
n
L )
R
RL
1 L
n⇡x
1 L
n⇡x
where an = L L f (x) cos( L ) dx and bn = L L f (x) sin( L ) dx
Fourier, sine and cosine series