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Introduction to PDE - INTRODUCTION To PBRTIBL DIFFERENTIAL...

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Unformatted text preview: INTRODUCTION To PBRTIBL DIFFERENTIAL 5000mm 0. PHRTML DIFFERENTIHL 5000170“) rm: DEFINITIDN LS smPLE awausu : LEE) :5 an 542123.170») 77> BE SoLvED FDR nu wxmm chnon: aF Two are noRE vlamfiBLES W wuoulas TmS Fumcnoro flND ITS PDRTmL DERWfifldfiS _ 5.6., FIND flu. £097) SBTISFyWG 3.5—+e-‘-—‘-:O 3X 3y LSEE limo 1713on smunoms you Cfln) FIND " By wsPEcnmo .. ) 7m: sway or sac» EQuBTmNS :5 ENORDAJS’L)’ CDHPMCIB'TED‘, HMEVERJ am) out? LOILL say JUST 3 F500 muss new? Sane mew anmnmnr manners. I. L f Dramas-010.91. wove Eauenom ) (20:95:ka 6 mm), smsnc smwc m9? :5 nanny 513570351, awn mam ”5L0 FIXED m ITS ENDPmMTS (.56., I9 GDITHE supp“; 3. 557,7,” 17077010 50 mm IT EXPERIENCES snfiLL, VERDceL wsmanoms ’ “ maxim" m TIDE If. .' amt) = was»; 0F me STRNG m wanna») x flmn me 1: 'PHVSICS onPuES Aug/t) , SBTLSFIES TH‘E I—DIflENSIoNBL wnvE gamma“, (42271525 c. :5 A comsmmr rum DEPENDS 0N com mismms Is nfiDE 0F 19:99 pm) nanny :7 1.5 smercusb L DEMSWI 3ND WNW” J. l I» TQITIVELYJ TUE VERTIcflL-flcctflskfifim L 953,: ) :5 PROPoRnDNBL 7” 2. THE CamcnUITy L 3'55) (finauru‘r 0F " BENDMJG" ) a EXERCISE 1 :_ 83000 777737 217::1: .211): SHU T 14091: ) : cos IS A 501.077010. 771$») DETERnnuE How THE STRWG NUST BE SET In) none» TD nfiKE :7” nova 77m MW, 0 IE. FIND we "'uvmm, mstcEnENT 11,090) 1 HAND " nomm. VELOCITY " gag-up), rwnuy, DESCRIBE (CfiREFULLy) How 7775 smwé 1.5 acmfiuy nomads C THE "0110» Is canes a smwmmc wnvE ), ’iL’iT’ THWH {98007 mi " SNBPSHOZS ” . E715 ! WIPE :5 01.30 a .3} D)DEN$IDN3L wave Banana» mm ozsémsz’s m. wBrmnoms a; 9 nEnBRfiNE L "Dunn ") onto .9 3- mnemmmm. wane EQUBJJOA) mnsmeo, Fan ExanPLE, ray mi COUPONENTS 0F 19:0 ELECTRonaewey-Ic FIELD, .1. L :_- throswmm. HEBT Eaunnam ) comsmiER a mu, npnpwwms we LYING Mme ms ram mamas» x=a man “I", O hm--- -_____ > X h I 2 I 3 X PRDWDE TUE «NEE “um 301715 INITIHL TEDPERBTDRE DASTRIBUHOM ( E. a. IMLD 19 mamas swam Ir ) 0ND WEN 350995 ”E ”597' J SOURCE . WERnaL masks)! Flows w Au fiTmnPT 72) acmfiwfi WEEDRL EQUIL‘13fi100 . >”’“"; 15 ) = TEUPERBWIPE BT Lac/9mm) X en») 7:05 1 PIHSICS If‘H’LIES #cxflt) SHTISFIES m I— DIUENSIDNHL #567 5003170» wHERE c :5 fl ComsmMT DETEPDINED By THE nammt at: «owe» we «come IS «600130550, EXERCISE «=2 : SHOW WT ’clex/I’el x #0917) = e. Cos "1;- L5 9 SOLUTION. DESCRIBE 771E [M11731 751713533ng ”1873980917010 Amp) fis men. as we torus 7mm Lt +°a ) Bahama? a? 77713 TEDPERAWHE Fumcnom . sons TERmrbaLosy 8ND mamnon : ms LfiPlfiCIflN ‘VJ‘ l 1 . ‘ . , .1 31.44 3.44 34! s _ ‘ I g . , lo .. g. +__— + .._—— a OMENS, mum. Lamacmm . v .44, - ax; 9 L 33; D nsnsm fl 3 4 .2 + ‘1 " ’ : : 4. NLLBPLBCDN V4 EL 9‘ . ,2 ’ I- DmENSIomm Minnow” : V a. = 3;; WHUE EQUBDOM.’ Z a 4 - 9.1.". «Hffi‘f 5009 770M! | an 3, (mmfici 5009mm ) “infra A. Dofis Nor DEFEND on t 8077/ m5 wBUE EOUBDON awn 771i ”597 540911040 REDUCE 1D 7711:: LHPLBCE EQufinom Vzuzo HERE ME 9 Few 0F me PHYSICBL SITUH’TIONS DESCRIBED B)’ WE LOP). 366 ERUB'WON .' STE/any smrg c 1.5., 7mg INDEPENDENT) mnPEkawa DISTRIBUDQNS E. 6., cawswER a 7m» Aeneas/v50” @138 OF nfiTERIBL ocCuPYIrDG a 25310» R 0F wf xy~,Pl.fimE, ~ y, 1 ! l a I l I l I t u‘..-------------‘ x a I :r 1m: sum :5 w 1715mm. saunmmun c mapmmugzs w ”or canines) fin») away) : VEDPEImmgg fiT « Locanm 09)!) ms» Jammy) SflflSFIE$ V144 = 0. a; RUBBER mm mm ITS waooky Is H mm: m wmcb’ 771:? SHEET :5 firearm», BEND m «ME 80 111197 Each POINT ts DISPLBGED VERDCBLLy 800E msmmce mews on 351.0“: m XY- Puma. 771E nEnBBfiNE I5 mow H 'GURUED suRFaas no SPficE fawn IF flex”) : ”EIGHT OF THE flEnBJi’flNEfiT (my) WM 44.. SfiflSFIES c. GRQVImnoNI-JL 0R swamasmnc P01510713). qucnoms Nona : Via. =0 vs GLso cauED ms POTENDBL 5003170” HND rrs SOLUTION; fliers ammo HBRnoMc mwcnoms, THE PROBLEFI 0F ”Granny Summing THESE EQUB‘naNS IS mo SJnPLE DB‘TTER AM: was «JILL DESCRIBE can): we tar-‘77:: nosr’r'BBmc 101519.; (9ND am! Fan THE I—Dmemsmmm. wave EQUATION, a, we u’msnaam 801.0119» .2. SEPfiRfiflDA) 0F vamaews D’HLEnEEnT's sownou :0 7E5 WWI-,3 5041mm) 6 m 2M'_ 9 ~ 3A, 3A“. ax’- ' 2x “WM” ' ’5; * ax : fifl+§firafi + ”Way-3 +32% M X 300' EX .2.— 3)‘ a 9X : 95¢ .214. 2.1!."- .592: 30'" + 3m)nl’ + 41’9” + w'L - 522 9‘» 2.9:. ' M" +4 3J9”? 9 1' 1. 97:44 2. 3‘4 _ ‘32:... 2—5-1) assets: 3': 8H0“) WM 3;“: = C C 24‘ '1 awn" 9 mos) ms wave 5403mm BECJflES 7. 3M. ___ 0 DUB» IMMRBTE wn’d RESPECT ID ru- TD 6157' .33}. = 11L“) ( Mar) M’BIT‘RAR)’ ) a»: JNTEGRBTE wmi RESPECT TD M 10 GET 2“. 1: SLIN)JMV 4- ¢L~r) 3w 3mm!“ :5 mu nm‘mmy Powwow Wow) OF M:- so 1,1,: ¢uxr) + Wild“). Magi): ¢Lx+c£)+4’Lx—c1:) ’ z EXEWCISE LI : L sows x x3), =- y 3)» 3y nan-M6 1m: 60131065 0F unmana- 31A all“. 3:74 ‘ .2 SOLVE 9x1 4» .1 ”any + 3);; 3y masons we Gimmes aF VMmeLEj g“: X M = X -y THIS PROCEDURE smPLE 9m) macaw-r, BUT NOT PBRTicoLmn)’ USEFUL w PRHCTICBL SI TUB DUNS . we mu. mow DEscmBE (we FIRST STEPS 0F ) n mcuma’us 772m :5 mi? 88515 F09? smog? flu. memes). CBchmnoMS. HERE :5 THE GENERBL PROBLEn we wont!) LIKE 7? saw: : C7- i341. = 2:; ) 0‘)! ‘2) 1. >0 LI-DIDENSMNHL wfiUE EQII) amt) :4“); t) = o J two c woman-:3 am; men) MLX‘O) ; put) , oexgfl L NWT/BL DisPLHCEDENT p00) %%‘x'°) ; o ) o e. x 5 g L wmm. VELocny ZERO) more 3 Fan Tut“ TmE BEING) “ME "0mm, DISPLflcEDEMT 0F ”E STRING cam 86 Hwy Fomcmm 9px) ow tin-,3] La”?! Pia) = pt?) =10. Bgcw By Loom—05 F01? sauna/vs TO 7771: FIRST 7100 PfiRK 0F 777E PROBLED ssmrmnmo 0F VBR’HBLES : as 0 60555 ( "amse'rzU 005 mm. Low! Fat? Sownons 0F W F03” #0926) = XcmTft’ IO. n, .314. , a,“ nawvanmu: 3; = XL») Tut) mun 3;; = me’w) 80 m P17077191. oemumwfi: 5R5 N000 anowfiky Dfimvnnués mvo we Haws 5005 firm a): Tum/m6 17/5 PDE meow wine» we mow wormm; 3 W7» sane ODE 5 Laeaurwmcu we ”New 8 LITTLE ), C I. 9x 313’” \ 7775 LEFT-”1WD 5105 :5 fl Fumcno» 0“ x. flND ms Piéfi7~HflND s»): A: :9 Manon 0F t , TITE amy way my MN 85 5430491. rs n= sac” Is Sane CONSTBNT 0" 50 II Xox) - O‘XLX) : 0 Tuct) - o-c" Tot) = 0 PM» of me Is I) caNSTbNT COEFFICIENT Eavenm 3:00 So 5037 m gave. WEI?) 4440,46) = Md, 16) = 0 lnPLIES X10) Toe) = XI!) Tat) =0 MR rm. 4% So Xlo) = X1£)=o LOWJMISE, Tot) s 0 so .ucx’t) s o meH IS 13 SOLUTION JD DIE «#105 EGQBDDN’ BUT BMW'OG ), F1857 more 77219715 { x"- O'X :0 X£0)= X4) =0 L5 3? ma nmtfiko SOLUTIADNSJ THEM 9: mm ’15 NEGBDUE , .I. wPPasE a-=0. WEN X" :0 so X00: 3,3141 50 ch) = 0 ==> £1. :0 ”ND X00 =4}. 307 mu; Xl1)=»0 =9 4'! :0 =3» 3 :0 so Xex) =0 11. SUPPOSE 0- >0) say) 0-: #1 Fm? sanéfi>o, 7715" um -nx x”-/fx =0 => Xac) = 45,.2 +32; LCWRacmmsnc saw/anon IS r‘. "‘20” rzift ) Xca)=0 =3; 43$; :0 So fibz’g. RN?) - x ch)= $.cem-e“ ), 77m) XI£>>0 =3» -1 .Q,LeM—8M )=O £1L61fi1°’ )=0 swag Ado moo .Q‘r‘o ms mmes 1%, so $0 «$1 "~‘3. =0 mafiamm, ch) = o, THUS) <1- noST 3E NEGBnnE 3ND wE tom. math? IT 135 o-=-).a‘ F91? Sons REEL NUI‘IBE'? A >0, was) owe EQDfinoNS BEConE u 4 x +xx:o cxm)=xw)=o) » L l - T + c A T ~ 0 amass 5' : sHow 777137 7711: sauna» 779 THE FIRST mum-mm :5 Xox) : AcosAx +Bsw>m mun Tim Xw) :19 =é l9=0 50 ch) : B SNAX Now) XtQ) = o ”muss BswAfi = 0. Suva: we CflN'TBLLOLO 3 m 35 25120 (FOR #715» ch) 50 mm amt) =0) we nusr m5 Sunk.“ = ,0 1,5. L2: n11“. F01? sons n=- 52., .-- ) mus, we omy Patemmew muss as A me me mags“ 017' ”r- '2' ) n"’L'-‘-- THESE flRE CHLLED THE EIGENVBLUES F0]? on): P1203151). FOR EM»- 0F 777E515 Lufi 03mm l9 801.0770» chx) : Sin Anx 1 SM) -—- TD x"+ ij :0 1.01171 ann Xnt-(7)= O. :3, Fox TUESE mamas 0F A m: T- Mormon BEconES sxaecrse é : 5mm 17787; Fax 51M» 0:), 2. ... ’ 'mE GENEML Seaman) .IS {numb natal 77,117): HnCOS 1 +806.M 1 warms fin moo 80 age manna}! comsmwrs. mus} Fate Er’qu n= 1,!) --- out? can) BUILD Sownaus uncx1£)=chxfl;¢g) 772 ms wave savanna» sfiTISFYWG flaunt) = Audit) =0 .- , t. wand. Mount) = swig—3‘- (Ha cos ”:2" + 308!» £ ) 34.. EXERCISE 3') : Snow mm 32- Lxm) =0 woman. vacnry 25kg) Recaps; mar 80 = 0 5a WESE 08E CULLED THE EIGEN Fumc‘noms OF OUR PROBLEn BM) THEY I’LL M 50775FY 3 OF THE H REQuiRED Cbnmrnaws LLflCmmG oNLy duty) = PM) ). 506» 0F 771555 FUNCTIONS REPRESENE B POSSIBLE DDTION OF 7715 mugs. LET'S SEE wvm wmm. DtsPLMEnBNTE GIVE PM; Tb sum! 00170105 n.5, FOR wmc» Pu) we IffivE coo...) sousb 00R PROBLEI') chPLETEL)’ ). FDR SmPucnyJ we mm. mug A" z I, M x t ) SIN nVTX‘ C05 tract L = -—— n J ‘2 1 :fi} n13)! Mango; = 6m; -€ were» 9 F5» or: mass wmm. DISPanEnEMTS : 12:0 n-I 11X 11.050): 6h) '2 No : F08 sac» to > 0 mi msmocsnebv dang/to 3 Is JUSTB WWPLE (87 C05 ”d" ) 0" “Emmet, Dasmcsnenr. Tires: 3R5 .smnmws .e‘ waves , 7776 11,, Out) BEE GfiLLED 'n-rE MORnflL 310355 OF Wanenow 0;: THE SWING, n: n I5 mt FIRST minnow: we FUNDI-DDENWJL no»; 3 , mee n 7 I DRE me HIGHfiR maknomcs coR ovate-Tomes ). EXERCISE 9 : FIND Tm: SOLUTION )9 THE wwmnmg 5779mm. 11,70,315” l c4 3-1" = 34‘- 04“.? in: ax], 211.- .i 3 Meat) =41fi£}=0, 17,0 X Ange) .- gnu '33..— ) o exsfl 5914-" - o X‘fl 31 (X10) —~ 0 , - - :9? 77m MINT “15 019M on»)! .50sz TUE. WERBWNG 37mm; PRoBLEn cob-EN 7715 mmm mspmcrsmzmr HfiPPENS 7298:? OF 771: Harm 77X Put) = Sling? ,osx5.€‘ mm) 77115 ISN’T nucH. HERE 15 $0111? HELP ; SUPERPOSITIDM PRINCIPLE : SUPPOSE magi): ..., An‘xat) rm. sansF-y ax“ ‘ M‘- xuoj) 2 14(1th : O 35%“,0) = 0 ”ND 19..-», 3%,), ms mam. munnms, 77713“) n Mug!) a 9-“.4"1t)*"'*3n4n0¢,t1 a 4’: I94 uflx’t) =1 (31-59 SfiTISFIES I’LL 777355, '6. T7715 REHSOA) ISVSIDPLE : 3191,“ 3'- C 5;: = 61511 (H,M,+--~+/9fi[email protected] ) 8‘4 51‘“ :- l ——----: + --+ “j, c (‘9' 5»: an War ) 9‘ Y 31.4 : ,4 (cl ___I + __ ,9 2. ‘n I 93“ ) + J) (0 Maxi ) 3314. 8211‘ : _—' "‘ , :4” 697‘ = 3;; L0.M'+-c—+0n¢“n) _ am an Mat) = 19.4.0,“ +~~~T an 14,, (at) : fi,-o +~~»+fin~0 : a Ml‘fJ‘t): B,M,££,.t)4w+ 490.3,, 0&3) a five) +v~-+fl_n'0 ‘ O a—‘icxo - 19 19311an3+0419 Eflfléxg)_.,a.o§m§8niozo a.‘ ' )- ’ 9‘ a sh M ‘4~ ; (,4 mam NOTICE WT THE Fumcnaw n Mama?) = Z: HQV“Q¢"~J5) =5 ” m I’M-rot i=3 27, 50 cat can Now sow; mt VIBIPBWNé smws Woman CanPLerz—y WHENEUER Tms INITIBL DISPLDCEDENT :5 OF WE FOR!) 113: 417x n17)! pLx) = )9, 3m) «6 + /9_z SIM,"'f +---+ fin 5110:6- . EXERCISE 9 .' SOLVE TUE VIBRHDNE. STRING PROBLEU 31.“ 3‘4 ‘7' 3;; ~— 31'" , Oleé'J 1’00 M£0,.t):ut.5;.£) :o ) two m-r 317x aux AUX = .——— —— ....... as. s ,0) 5 SM) 6’ +12 5m) 5 J X 5 344 3217901‘0 ’ péxéi THE NEXT STEP CWH!CH an? an“, NOT “MI-(E BECfiusE IT Esaumss A Pam flnovmr oi: won-4‘ ) :s ‘m 1?»?me THE Flmms LINE/9R ConBlmfiThfi’N 0F Eiéfimmwcflms '1 Em: C flwct Z 34?. sw T as «Q - :I 8y mo INFINITE 553mg °° @111 cht Mug/t) = Z: 94 5"" J1 80$ ,9 «RM ASSUDWB mm m). OF THE TEchmtcaL ISSUES CcoNVERGENcEw ETC‘ J can) BE 05:92.7 wma (THEY MN) 77115 DESCRIBE; 7715. I3 mono» as: THE smtmg IF WE anIfiL :‘DjsrvPLfiCEDENT IS on purrx «(/chJOJ =' Z ’25 5"” T , fl}; firvn Maw F9}? 7211? SURPRISE , 1715 Sfiaww IN 3725 735012] 0F FOURiER smwzs” nmr vmmauy may "REBSONDBIE " Fumcnpm Pm) emu BE 14))?th in; ma me pLX) = g [9% 81M m =4 So 7715 vwflarwcs STRING PROBLED I5 Basmmzy wwm Fox any INITH3L DISPLHCEDENT. EXERCISE )0 : n mm) smmam' wma LIES mam; ma wmevm. o a x51 0F WE x—mus THE-Win31. TEnPEEBTuRE DISTRIBUDON W 77778 wifi’E ’5 GWEN BY 771i? FUNCTION Path 05 x5 f. 7715 smug 9; THE (WEE mes HELD BT mnPERfiTURE 0, IF ungt) I5 7775 TEDPEIMTDRE or» 77-h? mIRE n’r LOCR‘DON x. 0ND TmE to THEN zaacngfi C 5;"M Jozx4£Jt>o 1410,!) :Aw’t)2a’ 25>]0 Mme» = Pm) . 02-sz ”I “”535 C1 :5 m5 ”7715mm. Din-"umww "AF 7775 “WE "9 WWW”) (a) Lb) LC) SEPBRBIE vBRMBLES 6ND $094077»)? 77715 omzy PDSSIRLE VBLUES 0;: 77m SEPBRHTMN comsmNT lemé’ 100.017?me Sozbnoms ME '1 Z. _. Viv 49”,” )1 J"‘ 53014.) WT HLL DF 7715 FDNCNDNS «Q‘rr‘th/g‘ .166, amt) : a 3m) Linc , 4% 4 «E =52 -.- .3 SBTMFY THE FIRST flop Palm; 0? WE Piflflbfl? flND osmfimme tome!) mmm TEDPERfin’PE MES 3’35 7?’ W TEDREIPBJDRE Evownom “g £fo 1'. FROM? 9 supskposmou Fremont; FOR 7715 H1367 cowoucnom PROBLEF) 6ND Descmsa m1. 0F 77:? wmm. fisnPERfiWRE D'STRIBoDows Pbx) Fm? coma-H 7m: P8081517 0610 BE SowED B)’ mess SUPERPas-moms, EXERCISE II : DissemBE 37 1.5657 T100 PHYSICBL SITua-noms GDUERNED By 9724 95.4 3xL+‘5;'L=0 ,04X4a_104)/<b Mm”): ML¢,7)=0 , Oéyéb MLx)b)= O )043‘5‘“ Qt ...
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