Separationof Variables
1 1 INTRO to PDEs
We firstintroduce somecanonicalPDEs that commonly occur in applications
I LAPLACE'S EQUATION Tu D
and therelated Poissonequation Dou f
Here D is called theLaplacianoperatorand canbewritten as DEA
In 2DCartesiancoordinates tu Egypt Uxxtuyy TautTyga
Fyn
In 3D Cartesiancoordinates O'u uxxtuyytu.az
ii HEATDIFFUSION EQUATION LOU
DI
used to describemanysituations in which a variablediffuses
The constant d is called thediffusivity andherewewilltypicallyscale d s
e heattransport on aplate
g
Iii WAVE EQUATION Tu
GIL c
denotes wavelikemotionof a variable u withconstant wavespeed c here wewilltypicallyset c s
iv HELMHOLTZEQUATION But Kau D
Forsome parameter KEIR
This is a timeinvariant form thewaveequationi e what is left youlookforsolutionsof
of if
the wave equation of the form u e't ie x y z
1 2 COORDINATE SYSTEMS
Wewill often work in other coordinatesystems The Laplacianhas differentformsin differentaerolinat
D POLAR COORDINATES u uCr O
x raso Tu
y r sino
f r
of tf guy I rur rt
igloo
ii CYLINDRICAL POLAR COORDINATES u air O Z
x raso r ur
rsino
U
F r
tf ugot azz
y
z z
iii SPHERICAL COORDINATES O E LO it
I
x rasosino O E LO21T
yz rsindsind
raso
XL
ft y
ou Fa rur tha waythanfindno UrrtYurtregroupthenno
r i
tf too
, You can derivethese from Cartesianexpressions
bychangingvariables
Or you can use different forms thegradient operator D in each coordinatesystem and
of derivatives
calculate D Ou taking track ofbasisvectors
of O EXERCISE BEYONDCOURSE
e.g in polers D Er or t to but for example geo
Eq
1 3 METHOD of SEPARATION of VARIABLES
IDEA find solutionsto PDEs in theformof a series ofseparatedsolutions
separated or separable solutions are simplythe product
of functions of one variableonly
eg is separable
f xyz is notseparable
f xtytz
EXAMPLE 1
Consider the ID Heatequation Ut lexx with oexe L L o t o
Find all nontrivial i e non zero separable solutions
Separable solution Ux t Xix Tct forsome function X and T
In the PDE
XII 232T or
Tt
LHS is a function of t only andRHS is a function of x only
Only possible bothsides are constants X forsome constant Xer
if I
Differentsolutionscanbegenerated depending on signof X
a d o X XX T XT
Acosta
X BSinha Celt someA B C
W XT Acoshrax Bsinhrxett A AC B BC
b 40 X D T D
X Ax B T C
W AxtB
Cc Xco set me d m o X MX T ut
d d
X AsinFux BasFux T Cent
U AsinfuxtBasfx e ut
applyforanytoo arm 0 However PDEs are typically furnished with boundaryconditions
Solutions
which place constraints on thecases cat b or co aboveand onthespecificvalues of that are term
admissable
EXAMPLE 2
Asabove butwithboundaryconditions acx o Ux L o ft
Revisitthe 3cases
a d so u Acoshrax Bsinhrxett
BC AO
BEETLE O B o
1 1 INTRO to PDEs
We firstintroduce somecanonicalPDEs that commonly occur in applications
I LAPLACE'S EQUATION Tu D
and therelated Poissonequation Dou f
Here D is called theLaplacianoperatorand canbewritten as DEA
In 2DCartesiancoordinates tu Egypt Uxxtuyy TautTyga
Fyn
In 3D Cartesiancoordinates O'u uxxtuyytu.az
ii HEATDIFFUSION EQUATION LOU
DI
used to describemanysituations in which a variablediffuses
The constant d is called thediffusivity andherewewilltypicallyscale d s
e heattransport on aplate
g
Iii WAVE EQUATION Tu
GIL c
denotes wavelikemotionof a variable u withconstant wavespeed c here wewilltypicallyset c s
iv HELMHOLTZEQUATION But Kau D
Forsome parameter KEIR
This is a timeinvariant form thewaveequationi e what is left youlookforsolutionsof
of if
the wave equation of the form u e't ie x y z
1 2 COORDINATE SYSTEMS
Wewill often work in other coordinatesystems The Laplacianhas differentformsin differentaerolinat
D POLAR COORDINATES u uCr O
x raso Tu
y r sino
f r
of tf guy I rur rt
igloo
ii CYLINDRICAL POLAR COORDINATES u air O Z
x raso r ur
rsino
U
F r
tf ugot azz
y
z z
iii SPHERICAL COORDINATES O E LO it
I
x rasosino O E LO21T
yz rsindsind
raso
XL
ft y
ou Fa rur tha waythanfindno UrrtYurtregroupthenno
r i
tf too
, You can derivethese from Cartesianexpressions
bychangingvariables
Or you can use different forms thegradient operator D in each coordinatesystem and
of derivatives
calculate D Ou taking track ofbasisvectors
of O EXERCISE BEYONDCOURSE
e.g in polers D Er or t to but for example geo
Eq
1 3 METHOD of SEPARATION of VARIABLES
IDEA find solutionsto PDEs in theformof a series ofseparatedsolutions
separated or separable solutions are simplythe product
of functions of one variableonly
eg is separable
f xyz is notseparable
f xtytz
EXAMPLE 1
Consider the ID Heatequation Ut lexx with oexe L L o t o
Find all nontrivial i e non zero separable solutions
Separable solution Ux t Xix Tct forsome function X and T
In the PDE
XII 232T or
Tt
LHS is a function of t only andRHS is a function of x only
Only possible bothsides are constants X forsome constant Xer
if I
Differentsolutionscanbegenerated depending on signof X
a d o X XX T XT
Acosta
X BSinha Celt someA B C
W XT Acoshrax Bsinhrxett A AC B BC
b 40 X D T D
X Ax B T C
W AxtB
Cc Xco set me d m o X MX T ut
d d
X AsinFux BasFux T Cent
U AsinfuxtBasfx e ut
applyforanytoo arm 0 However PDEs are typically furnished with boundaryconditions
Solutions
which place constraints on thecases cat b or co aboveand onthespecificvalues of that are term
admissable
EXAMPLE 2
Asabove butwithboundaryconditions acx o Ux L o ft
Revisitthe 3cases
a d so u Acoshrax Bsinhrxett
BC AO
BEETLE O B o
