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Math 126 Midterm Exam Spring 2020
Name: Student ID #:This exam has 10 pages, 4 questions, and a total of 100 points.
1. Assume u : R2 ! R is C2.
(a) (10 points) Solve
utt = uxx for x 2 R and t > 0 where
(u(x, 0) = e�x, x 2 R
ut(x, 0) = 0, x 2 R.
(b) (10 points) Solve
utt = uxx, for x > 0 and t > 0
8><
>:
u(0, t) = 0, t > 0
u(x, 0) = e�x, x > 0
ut(x, 0) = 0, x > 0.
Ryan's
x ex
4 x OBy d Alembert's formula
Xt
II If 1for X EIR t 0
x ex
4 x O
By d Alembert's formula xtU x t 12 0 xtt t x 1 0Ily
t Xt
Lz et
Iz et for c o tand X it
U x t z x it 112 0 x t tfodyX t
Lz eX E Iz e
tfor c too
Dong Gyu
(c) (15 points) Suppose u : R2 ! R is C2. Find the general solution of
utt + uxt � 12uxx = 0
and show it satisfies the PDE.
Page 2
HereL u3 2ttu 2t u 122 4
and by Clairaut's Thuru 42 4 32 4 122 4
2e 142 712 32 7 u O
Then
Z 142 14 0 or Z 32 7 u 0
are transport equations wherex 4th and g x 13T
form the general solutionThus
ucx.tt f x 4T g x 13T
is the general solution where f g E C IR
Note
Utt 167 X 14T 9g X 3TU t 4 f X 14T 3g X 3T
12U xx 127 X 14T 12g X 3T
O
2. Consider the equation(y + u)ux + yuy = x� y
where u : R2 ! R is C1.
(a) (10 points) Given the initial data
u(x, 1) = 1 + x
find the characteristics.
Page 3
Our system of ODES is
Xp 4 tu X S o s
4T y y Cs 01 1
Up X y u S o Its
NOWXyz yet he y X y X
and soX STL aisle 4 bcs e t
X S O at b s
X45,01 415,01 1 UCS01 from the system2tsa b from the general sold
a I s b I
X S T Its et eT
Next
He _y yes 4 etand our characteristic curves are
X its et Let
Its y 44
(b) (10 points) Given the initial data
u(x, 1) = 1 + x
find the explicit solution for u or explain why none can exist.
(c) (5 points) Give an example of a connected curve C in R2 such that the PDE with pre-scribed data on that curve cannot be solved.
Page 4
From above
UT set et
ucs 4 Sette T
Solving for S and T yieldsX its y Yy s I Xyy Yy l4
andy Intl
Henceu x y y
t ly 1 4 t ly
x y I4
choose C to be a characteristic let 5 0
ix y I X Holy lyor a curve that never intersects a characteristic
3. (20 points) Show that for
u = f⇣xt
⌘
to be a nonconstant solution ofut + a(u)ux = 0
then f must be the inverse function of a.
Page 5
Idea : Play an ufxit) = HI) into the equation and use Chain Rule appropriately.UxCatt = offCI)) = I - f 'CI) (chain Rule) .At Gutt = ITCHED = -¥ - f'f¥) ( n )
.
Now,if you plug them into the given equation Ut take)UFO
, you get- EE f ' + ACHED . If
'
= o.
We can rewrite this as
- (¥ - afff#D. If =o.
So,there are two cases : f-
'
(¥) -0 OR ACHED = It .
However,
this holds
for any real numbers K & t - So,
we can see what happens at A- At where
k is your favorite number . Now,for any DEIR
,we should have
f-'(d) =o OR afffd)) =D .
But , Tu fact , if the product of two antiheroes functions Ts O,then
one of them should be the zero function.
In this problem ,two functions are fld) & aHGH) - d .
But, we know that
U is not constant which empties that flat is not constantly 0 .
Hence. afffd)) - to.
Therefore,f is the inverse function of a .
4 It is necessary to check that alf =D for all AER.
4. In lecture we proved the following:
Theorem: Suppose that u1 and u2 solve the IVP utt = c2uxx with displacement functions �1
and �2 and velocity functions 1 and 2 respectively.
Thenk�1 � �2k1 < ✏ and k 1 � 2k1 < ✏) ku1 � u2k1 < (1 + t)✏
for t > 0.
(a) (5 points) State a similar result for the PDE
utt = c2uxx + f(x, t).
Page 6
[ This is It'not t'
if youdid not make a mistake on writing
the Duhamel solution .
(b) (15 points) Prove your result for (a) for c = 1.
Page 7
Idea : Use Duhamel solution and simply bound the difference using maximum values.
uicx.tt = I Hat to.fr#fezIfiiiituHadytzISotfaEiiEs9f.fy.ssdydsis the solution for each equation (i = I.2) .
By subtracting Us from u. .we get
lkfktt - Uda .tk If tutto.Cx-AB - I date) talk-ED
toe SEEING - E Siiiitukcsldgthe Sot fit fica .sidads - ztcfofiiiii
.
dads.
Hence,KlickH - adat)(s I lolcat CH -Beat#HI (Watt -Elk-CHI
txfiiiathhcyi -Waldy ta iii. Head-fiscaldads
Here,we used the triangle inequality and its integral version . namely, I Eff E Sab Ifl
.
Recalling the definition Hellas = sup lol , we get
z' lol ,Heat - oblate)k I Hol .- oldto LIE
I told'-CH - oldx -attEI Kok-Elks < IE
Ic fi Hast -rkcyildy c- Ec SITE Huh-441 dy CE . set - E
and finally+ . ginleyratingat ft if HIGH - fdassldydss# E' fit Hfifzlfsdyds this region.
< Loft Siiiiiiiiiedyds act a met
= # ft zaf-sleds = ZICEE- EEE. .
Adding them up . we obtain thatkhat - adatK Ette t EEE = (Htt⇒E for any AER .
Therefore, Hui-Ulla : = asueffz (UHH -Udaithe (Htt EE)E .
{ To avoid the issue of getting € instead of <,you need to be a bit careful .
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