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Chapter6PartialDifferentialEquationsI

PHYS4840Prof.HannahJang-Condell

Announcements

• Problemset5dueWednesday,April12• Noclass4/20,noofficehours4/21• Problemset6dueWednesday,April26• Thefinalproblemsetisoptional,andwilltaketheplaceofyourcurrentlowestproblemsetscore.

• ThefinalexamwillbeThursday,May11at10:15amin**ENG2105**

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OrdinaryDifferentialEquationsvs.

PartialDifferentialEquations

∂u∂t, ∂

2u∂t2,

∂u∂x, ∂

2u∂x2,

∂u∂y, ∂

2u∂y2,

dxdt, d

2xdt2,

Generalsecond-orderPDE

• Hyperbolic• Parabolic• Elliptic

A∂2u∂x2

+ B ∂2u∂x∂y

+C ∂2u∂y2

+D ∂u∂x

+ E ∂u∂y

+ Fu(x, y)+G = 0

B2−4AC > 0B2−4AC = 0B2−4AC < 0

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PDEExamples

• Hyperbolic(Ch 7)–Waveequation(7.1)

• Parabolic(Ch 6)– Diffusionequation(6.2)

• Elliptic(Ch 8)– Poissonequation ∂2u

∂x2+∂2u∂y2

= ρ(x, y)

∂u∂t=∂∂x

D ∂u∂x

"

#$

%

&'

∂2u∂t2

= v2 ∂2u∂x2

Initialvalueproblems

vsBoundaryvalue

problems

imagecredit:NumericalRecipesinC

time

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Discussion

• Cantheboundaryconditionsforaninitialvalueproblembederivatives(andnotjustboundaryvalues)?

• Ifyouclaimthatderivativescouldbeused,whatkindofderivativeswouldtheybe:withrespecttotimeorspace?

Typesofboundaryconditions

• Dirichlet𝑢 𝑥 = 𝑎, 𝑡 =fixed

• Neumann𝑑𝑢𝑑𝑥 𝑥 = 𝑎, 𝑡 = 0

• Periodic(wrapsaround)

x

x=a x=b

𝑢 𝑥 = 𝑎, 𝑡 = 𝑢 𝑥 = 𝑏, 𝑡𝜕𝑇𝜕𝑥-./0

=𝜕𝑇𝜕𝑥-./1

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PeriodicboundaryconditionsSupposewearerunningasimulationon10classicalparticles,eachofmassmandinitiallyatrestwithinacubeofdimensionL.Furthersupposethatweareemployingperiodicboundaryconditions,onallfacesofthecube,forboththepositionandvelocityoftheparticles.Eachparticlehasaspeedsuchthatitmovesadistancedxinonetimestepdt.Oneoftheparticlesinoursimulationisperpendicularlyapproachingthecenterofthebottomsideofthecube;attimestepNtheparticleis0.5*dxabovethecenterofthebottom.Inoursimulation,attimestepN+1wheredowefindthisparticle?

0.5*dx v=dx/dt

L

Supposeyouhavealongmetalbarthatconductsheat.

• HowwouldyoucreateDirchlet boundaryconditions?

• HowwouldyoucreateNeumannboundaryconditions?

• Howwouldyoucreateperiodicboundaryconditions?

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• ParabolicandhyperbolicPDEsaretypicallyinitialvalueproblemswithsomeboundaryconditionsimposed.

• EllipticPDEsaretypicallyboundaryvalueproblems

Solvinganinitialvalueproblem

t=nτ

x=−L/2+ihi = 0, … , Nh = L/N

ui,n=u(xi , tn)

time

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Diffusion

• Supposeyouheatthecenterofaconductingmetalbartoahightemperature.TheendsofthebararefixedtoT=0 K.Howdoyouexpectthetemperatureasafunctionofpositionalongthebartoevolve?(Assumenoheatescapestotheairaroundit.)

Diffusion

• Example:heattransfer𝜕𝜕𝑡 𝑇 𝑥, 𝑡 = 𝜅

𝜕3

𝜕𝑥3 𝑇(𝑥, 𝑡)

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Diffusion

• Example:heattransfer𝜕𝜕𝑡 𝑇 𝑥, 𝑡 = 𝜅

𝜕3

𝜕𝑥3 𝑇(𝑥, 𝑡)

• Analyticsolution:

𝑇6 𝑥, 𝑡 =1

𝜎(𝑡) 2𝜋� exp− 𝑥 − 𝑥@ 3

2𝜎3 𝑡𝜎 𝑡 = 2𝜅𝑡�

• Ast→0,limD→@

𝑇6 𝑥, 𝑡 = 𝛿(𝑥 − 𝑥@)

Diffusionsolution

x

T

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Methodofimages

• Constructasolutionfromasuperpositionofstates

𝑇 𝑥, 𝑡 = G −1 H𝑇6(𝑥 + 𝑛𝐿, 𝑡)L

H/ML

• ValidforDirchlet boundaryconditions,x0=0

Methodofimages

• Dirchlet

x

T

–L/2 L/2 –L L

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Methodofimages

• Neumann

x

T

-L/2 L/2 –L L

Methodofimages

• Neumann

x

T

-L/2 L/2 –L L

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ForwardTimeCenteredSpaceScheme

• Howdoyoucalculatethetemperatureevolutioncomputationally?

• Forwardtimederivative𝜕𝜕𝑡 𝑇 𝑥, 𝑡 =

𝑇(𝑥N, 𝑡H + 𝜏) − 𝑇(𝑥, 𝑡H)𝜏 =

𝑇NHPQ − 𝑇NH

𝜏• Centeredspacederivative

𝜕3

𝜕𝑥3 𝑇 𝑥, 𝑡 =𝑇NPQH + 𝑇NMQH − 2𝑇NH

ℎ3

FTCS

𝑇NHPQ = 𝑇NH +𝜅𝜏ℎ3 𝑇NPQH + 𝑇NMQH − 2𝑇NH

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Stabilityvs.efficiency

• Initialvalueproblem:stabilitymatters• Boundaryvalueproblem:efficiencymatters

• Stabilityforthediffusionproblem:

isthemaximumtimestep

tσ =h2

2κ