laplace 's equation in a rectangle

10
Laplace 's Equation in a Rectangle

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Page 1: Laplace 's Equation in a Rectangle

Laplace 's Equationin a Rectangle

Page 2: Laplace 's Equation in a Rectangle

⇒ = 28%17 Equilibrium solutions

x g)iuakoasioasEe -- al:# FED+ BcsTo

what is the shape of ourheated -plate ?

Ffa (x)

i¥¥⇒¥⇒*÷. :÷÷÷÷.yeah fam , XEEOIKJ

Page 3: Laplace 's Equation in a Rectangle

Laplaces Equation in a rectangle .-

[0,k]x[o , if= Lcxiyxypi : xtfoikieefo.bg

Tuya t 8%2=0 teamX C- Cork]

Subject to Bcs @G,o)=falx),uGiD=f#Tucx.iq#Giy7--gelybmCKiy)=gzCy7T-T¥thisprohlanintEf"problemugjp-thcx.pt Uday)-this is the fol . of the original

problem .

Page 4: Laplace 's Equation in a Rectangle

Solving the orange problem

.us#ui%EEHseLEYoiIfyDPDEA4x7Bly7tA B''

Cy)=O H

Alo)BIyt-AIKJBHI-os-o.tt#i,=BEFy,--X④

Fearne .

TAa9E[ have eigenvalueproblem for A-

→ ELI)A=aAa¥¥Seen before :

Page 5: Laplace 's Equation in a Rectangle

.FI#z:eEmt:E.ntFsxHEIEsJThe equation for Bis fB.gs#I5Bcp--oTIIYoasEaaeeEpfoe.÷÷olutioasisA- calculus {eIpf7Iy) ,expGEy))gtndeutouedgioe##

Page 6: Laplace 's Equation in a Rectangle

A:c:u9aYh{ chatty)

.su#JBF*FaegEoooE:wukin--acxsBg,¥Y=¥¥Mlxio)=ACx> Blog

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Page 7: Laplace 's Equation in a Rectangle

Based on what we want I choose

the of solutions tobe si:i÷¥÷. . :i¥÷÷¥}Finally I have the productsolutionsmeIN sin9¥11:¥f;¥D , sinti shtick- sheetunit"" P mmr Kil) "

p

Page 8: Laplace 's Equation in a Rectangle

solution rig, bysuperposition

nkifzam.nmmnlxiptzampum.LK,y)'matIbring back f, Ife

f, = UH , 0352am.im#ntJam.qum.dxI--Zam.asiutiIx) 's ✓

am,z= ¥ Iof , sin (YI¥

Page 9: Laplace 's Equation in a Rectangle

f-zk7=uCxiL)=Iamµum, .HR#m.Um.zki4in

faK7=ZamnsiYTEx)⇒

am . --¥¥NsinCEx)dx-

makeup -_ Ibm.su/mIDshsfTEftgt+Zbm.zsiufEy)shhmIk.xDThet)bm.EE?gixsialmEDdy,bm..=ES..gaeDsintnEyIdy

Page 10: Laplace 's Equation in a Rectangle