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 .