Par
tial D
iffen
tialE
quat
ions
Man
y im
port
ant
phys
ical
pro
cess
es in
na
ture
are
gov
erne
d by
PD
Es.
To u
nder
stan
d th
e ph
ysic
al b
ehav
ior
of
the
mod
el r
epre
sent
ed b
y th
e PD
E is
im
port
ant.
Know
ledg
e of
the
mat
hem
atic
al c
hara
cter
, pr
oper
ties,
and
sol
utio
n of
the
PD
Esis
re
quire
d.
Phy
sica
l Cla
ssifi
catio
n
Equi
libriu
m P
robl
ems:
a g
iven
PD
E is
des
ired
in a
cl
osed
dom
ain
subj
ect
to a
pre
scrib
ed s
et o
f BC
s.St
eady
-sta
te t
empe
ratu
re d
istr
ibut
ions
, in
com
pres
sibl
e in
visc
id f
low
s, e
quili
briu
m s
tres
s di
strib
utio
ns in
sol
ids
(elli
ptic
PD
Es).
Ste
ady-
Sta
te T
empe
ratu
re
Dis
tribu
tions
Lapl
ace’
s eq
uatio
n
Sol
utio
n
Sepa
ratio
n of
var
iabl
es (
Gre
ensp
an, 1
961)
: T(
x, y
)=X(
x)Y(
y)If
a s
olut
ion
of t
his
form
can
be
foun
d th
at
satis
fies
both
the
PD
E an
d th
e BC
s: t
he
one
and
only
sol
utio
n to
the
pro
blem
(W
einb
erge
r, 1
965)
PDE
is r
educ
ed t
o tw
o O
DEs
.
Solu
tion
at a
ny p
oint
inte
rior
to t
he d
omai
n of
in
tere
st d
epen
ds u
pon
the
spec
ified
con
ditio
ns a
t al
l poi
nts
on t
he b
ound
ary.
Inco
mpr
essi
ble
Invi
scid
Flo
ws
Lapl
ace’
s eq
uatio
n
Sol
utio
n
Solv
ed b
y co
mbi
ning
tw
o el
emen
tary
sol
utio
ns
of L
apla
ce’s
eqn
. tha
t sa
tisfy
the
BCs
.Fo
r a
linea
r PD
E, a
ny li
near
com
bina
tion
of
solu
tions
is a
lso
a so
lutio
n (C
hurc
hill,
194
1).
Not
es
Equi
libriu
m P
robl
ems
are
refe
rred
to
as ju
ry
prob
lem
s.Th
e so
lutio
n of
the
PD
E at
eve
ry p
oint
in
the
dom
ain
D d
epen
ds u
pon
the
pres
crib
ed
BC a
t ev
ery
poin
t on
B.
Mar
chin
g Pr
oble
ms:
tra
nsie
nt o
r tr
ansi
ent-
like
prob
lem
s w
here
the
sol
utio
n of
a P
DE
is
requ
ired
on a
n op
en d
omai
n su
bjec
t to
a s
et o
f IC
s an
d Bc
s.Tr
ansi
ent
tem
pera
ture
dis
trib
utio
n (p
arab
olic
PD
Es),
dis
plac
emen
t of
a s
trin
g (h
yper
bolic
PD
Es)
Tran
sien
t Tem
pera
ture
Dis
tribu
tion
Hea
t eq
uatio
n
Sol
utio
nSe
para
tion
of v
aria
bles
: T(
x, t
)=u(
x)+
v(x,
t)St
eady
pro
blem
sol
.+ t
rans
ient
sol
. tha
t di
es o
ut a
t la
rge
times
.
Dis
plac
emen
t of a
Stri
ngW
ave
equa
tion
Sol
utio
n
Solu
tions
for
pro
blem
s of
thi
s ty
pe u
sual
ly
requ
ire a
n in
finite
ser
ies
to c
orre
ctly
app
roxi
mat
e th
e in
itial
dat
a.
Mat
hem
atic
al C
lass
ifica
tion
Mat
hem
atic
al c
once
pt o
f ch
arac
teris
tic li
nes
(2D
) or
sur
face
s (3
D):
alo
ng w
hich
cer
tain
pr
oper
ties
rem
ain
cons
tant
or
cert
ain
deriv
ativ
es m
ay b
e di
scon
tinuo
us.
Char
acte
ristic
line
s or
sur
face
s ar
e th
e di
rect
ions
whe
re “i
nfor
mat
ion”
can
be
tran
smitt
ed in
phy
sica
l pro
blem
s go
vern
ed
by P
DEs
.
2nd
orde
r lin
ear
PDE:
a, b
, c, d
, e, f
are
fun
ctio
ns o
f (x
,y)
only
.
(2.1
5a)
Clas
sific
atio
n of
a 2
nd o
rder
PD
E de
pend
s on
ly o
n th
e se
cond
der
ivat
ive
term
s of
the
eqn
.
The
char
acte
ristic
s, if
the
y ex
ist
and
are
real
cur
ves
with
in t
he s
olut
ion
dom
ain,
rep
rese
nt t
he lo
cus
of
poin
ts a
long
whi
ch t
he s
econ
d de
rivat
ives
may
not
be
con
tinuo
us.
Iden
tify
Cha
ract
eris
tics
Cur
ves
May
the
re b
e lo
catio
ns w
here
the
sec
ond
deriv
ativ
es a
re
disc
ontin
uous
?Le
t τ
be a
par
amet
er t
hat
varie
s al
ong
a cu
rve
C in
the
x-
y pl
ane.
Tha
t is
, on
C, x
=x(τ)
and
y=
y(τ)
. For
co
nven
ienc
e, o
n C,
we
defin
e
The
curv
es y
(x)
that
sat
isfy
abo
ve e
qnar
e ca
lled
the
char
acte
ristic
s of
the
PD
E.Al
ong
thes
e cu
rves
, the
sec
ond
deriv
ativ
es a
re
not
uniq
uely
det
erm
ined
by
spec
ified
Φ, Φ
x , Φ
y.(b
2 -4a
c) p
lays
a m
ajor
rol
e in
the
nat
ure
of t
he
char
acte
ristic
cur
ves.
(b2 -
4ac)
>0:
tw
o di
stin
ct f
amili
es o
f re
al
char
acte
ristic
cur
ves
exis
t (h
yper
bolic
PD
E)(b
2 -4a
c)=
0: s
ingl
e fa
mily
of
char
acte
ristic
cur
ves
exis
t (p
arab
olic
PD
E)(b
2 -4a
c)<
0: n
o re
al c
hara
cter
istic
cur
ves
exis
t (e
llipt
ic P
DE)
PD
E in
Cha
ract
eris
tic C
oord
inat
e Fo
rm:
(x, y
)→( ξ
, η) n
onsi
ngul
ar m
appi
ngH
yper
bolic
PD
E:
Φξξ
-Φηη
=h 1
(Φξ,
Φη,
Φ, ξ
, η)
Φξη
=h 2
(Φξ,
Φη,
Φ, ξ
, η)
Para
bolic
PD
E:Φ
ξξ=
h 3(Φ
ξ, Φ
η, Φ
, ξ, η
)or
Φηη
=h 4
(Φξ,
Φη,
Φ, ξ
, η)
Ellip
tic P
DE:
Φξξ
+Φ
ηη=
h 5(Φ
ξ, Φ
η, Φ
, ξ, η
)
We
are
assu
red
of a
non
sing
ular
map
ping
, pro
vide
d th
at
the
Jaco
bian
of
the
tran
sfor
mat
ion
is
no
nzer
o (T
aylo
r, 1
955)
.An
y re
al n
onsi
ngul
ar t
rans
form
atio
n do
es n
ot c
hang
e th
e ty
pe o
f PD
E.
Hyp
erbo
lic P
DE
s
Two
dist
inct
fam
ilies
of
char
acte
ristic
s ex
ist.
Seco
nd-o
rder
wav
e eq
uatio
n: u
tt=
c2u x
x
-∞<
x<+∞
, u(x
,0)=
f(x)
, ut(x
,0)=
g(x)
Sol
utio
nTr
ansf
orm
atio
n to
cha
ract
eris
tic c
oord
inat
es
perm
its s
impl
e in
tegr
atio
n of
the
wav
e eq
uatio
n:
u ξη=
0 w
here
ξ=
x+ct
, η=
x-ct
.
Firs
t te
rm:
prop
agat
ion
of t
he in
itial
dat
a al
ong
the
char
acte
ristic
sSe
cond
ter
m:
effe
ct o
f th
e da
ta w
ithin
the
clo
sed
inte
rval
at
t=0
Righ
t ru
nnin
g ch
arac
teris
tic:
+(1
/c)
Left
run
ning
cha
ract
eris
tic:
-(1/
c)
Fund
amen
tal P
rope
rty o
f H
yper
bolic
PD
ELi
mite
d do
mai
n of
dep
ende
nce
is b
ound
ed
by t
he c
hara
cter
istic
s th
at p
ass
thro
ugh
the
poin
t (x
0, t
0).
Any
dist
urba
nce
that
occ
urs
outs
ide
of t
his
inte
rval
can
nev
er in
fluen
ce t
he s
olut
ion
at
(x0,
t0)
.
Par
abol
ic P
DE
sPa
rabo
lic P
DEs
asso
ciat
ed w
ith d
iffus
ion
proc
esse
s.So
lutio
n of
par
abol
ic e
quat
ion
at t
ime
t 1de
pend
s up
on t
he e
ntire
phy
sica
l dom
ain
(t≤t
1), i
nclu
ding
an
y si
de B
Cs.
Ray
leig
hpr
oble
m:
2-D
fla
t pl
ate
impu
lsiv
ely
acce
lera
ted
to
a ve
loci
ty U
fro
m r
est.
Sim
ilarit
y so
lutio
n
(Han
sen,
196
4)
Gro
wth
of
this
laye
r is
con
trol
led
by ν
and
t.Ve
loci
ty c
hang
e in
the
laye
r is
indu
ced
by d
iffus
ion
of t
he
plat
e ve
loci
ty in
to t
he in
itial
ly u
ndis
turb
ed f
luid
.
Erro
r fu
nctio
n
Elli
ptic
PD
Es
The
disc
rimin
ant
is n
egat
ive
(b2 -
4ac<
0) a
nd
the
char
acte
ristic
diff
eren
tial e
quat
ion
has
no r
eal s
olut
ion.
It m
eans
tha
t th
ere
is n
o re
al c
hara
cter
istic
s ex
ist
and
it is
mea
ning
less
to
find
them
.Th
e so
lutio
n of
elli
ptic
PD
E de
pend
s up
on
cond
ition
s on
the
ent
ire b
ound
ary
of t
he
clos
ed d
omai
n.
Wel
l-Pos
ed P
robl
em
A un
ique
sol
utio
n to
PD
Esca
nnot
be
obta
ined
if t
he in
itial
dat
a or
bou
ndar
y co
nditi
ons
is im
prop
er u
sed.
A pr
oble
m in
volv
ing
a PD
E to
be
wel
l-po
sed,
the
sol
utio
n to
the
pro
blem
mus
t ex
ist,
mus
t be
uni
que,
and
mus
t de
pend
co
ntin
uous
ly u
pon
the
initi
al o
r bo
unda
ry
data
(Had
amar
d, 1
952)
.
Diff
eren
t Typ
es o
f BC
s
Diri
chle
tBC
s[u B
=f]
+ P
DEs D
⇒D
irich
let
Prob
lem
Neu
man
n BC
s[∂
u/∂n B
=f]
+PD
Es D
⇒N
eum
ann
Prob
lem
Rob
in’s
BCs
: [(∂u
/∂n+
ku)
B =
f] +
PD
Es D
⇒Rob
in’s
Pro
blem
u
Sys
tem
s of
Eqn
s.In
app
lyin
g nu
mer
ical
met
hods
to
phys
ical
pr
oble
ms,
sys
tem
s of
eqn
s. a
re f
requ
ently
en
coun
tere
d.Th
e pr
oces
s is
gov
erne
d by
a h
ighe
r-or
der
PDE,
th
e PD
E ca
n us
ually
be
conv
erte
d to
a s
yste
m o
f 1s
t or
der
equa
tions
.W
ave
equa
tion:
utt=
c2u x
x
⇒∂v
/∂t=
c∂w
/∂x,
∂w
/∂t=
c∂v/∂x
whe
re v
=∂u
/∂t,
w
=c∂
u/∂x
Lapl
ace
equa
tion:
uxx
+u y
y=0
⇒∂u
/∂x=
∂v/∂
y, ∂
u/∂y
=-∂
v/∂x
Sys
tem
of 1
st O
rder
Eqn
s.
Cons
ider
cur
ves
C on
whi
ch a
ll bu
t th
e hi
ghes
t or
der
deriv
ativ
es a
re s
peci
fied
and
inqu
ire a
bout
co
nditi
ons
that
will
indi
cate
tha
t th
e hi
ghes
t de
rivat
ives
are
not
uni
quel
y de
term
ined
.W
e le
t a
para
met
er τ
vary
alo
ng c
urve
s C
and
use
the
Chai
n ru
le t
o w
rite
the
follo
win
g Eq
ns.
Writ
ing
the
four
equ
atio
ns in
mat
rix f
orm
:
A un
ique
sol
utio
n fo
r th
e fir
st d
eriv
ativ
es o
f u(
x, y
) an
d v(
x, y
) do
es n
ot e
xist
if t
he d
eter
min
ant
of
the
coef
ficie
nt m
atrix
is z
ero.
Clas
sific
atio
n of
1st
ord
er s
yste
m:
D>
0: H
yper
bolic
sys
tem
, D=
0: P
arab
olic
sys
tem
, D
<0:
Elli
ptic
sys
tem
.
In 1
st o
rder
sys
tem
, if
the
root
s of
the
ch
arac
teris
tic e
qns.
are
con
tain
bot
h re
al
and
com
plex
par
ts, t
hen
the
syst
em is
m
ixed
and
may
exh
ibit
hype
rbol
ic, p
arab
olic
, an
d el
liptic
beh
avio
r.Th
e cl
assi
ficat
ion
of s
yste
ms
of 2
nd o
rder
PD
Esis
ver
y co
mpl
ex.
It is
diff
icul
t to
det
erm
ine
the
mat
hem
atic
al
beha
vior
of
thes
e sy
stem
s ex
cept
for
sim
ple
case
s.
Oth
er d
iffer
entia
l equ
atio
ns o
f int
eres
t
2nd
equa
tions
: W
ave
equa
tion,
Hea
t eq
uatio
n, L
apla
ce’s
equ
atio
n.A
num
ber
of o
ther
ver
y im
port
ant
equa
tions
sho
uld
be m
entio
ned,
sin
ce t
yey
gove
rn c
omm
on p
hysi
cal p
heno
men
a or
th
ey a
re u
sed
as s
impl
e m
odel
sfo
r m
ore
com
plex
pro
blem
s. I
n m
any
case
s, e
xact
an
alyt
ical
sol
utio
nsfo
r th
ese
equa
tions
ex
ist.
