This chapter introduces important PDEs such as the heat equation, wave equation, Laplace equation and Helmholtz equation and gives examples of how to solve them using separation of variables. Then it introduces the 3 coordinate systems of polars, cylindrical polars and spherical, showing how to go ...
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
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