Fourier Transform Methods for Partial Differential Equations

Transcription

1 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio,, Vol, No 3, -57 Avill oli t Si d Edutio Pulihig DOI:69/ijpd--3- Fourir Trorm Mthod or Prtil Dirtil Equtio Nol Tu Ngro * Dprtmt o Mthmti, Collg o Nturl d Computtiol Si, Wollg Uivrity, P o 395 Corrpodig uthor: tit@gmilom Rivd My 6, ; Rvid Ju, ; Aptd Ju 9, Atrt Th purpo o thi mir ppr i to itrodu th Fourir trorm mthod or prtil dirtil qutio Th itrodutio oti ll th poil ort to ilitt th udrtdig o Fourir trorm mthod or whih qulittiv thory i vill d lo om illutrtiv mpl w giv Th rultig Fourir trorm mp utio did o phyil p to utio did o th p o rqui, who vlu qutiy th mout o h priodi rquy otid i th origil utio th ivr Fourir trorm rotrut th origil utio rom it trorm Kyword: ourir trorm, prtil dirtil qutio Cit Thi Artil: Nol Tu Ngro, Fourir Trorm Mthod or Prtil Dirtil Equtio Itrtiol Jourl o Prtil Dirtil Equtio d Applitio, vol, o 3 (: -57 doi: 69/ijpd--3- Itrodutio Th Fourir trorm i th turl tio o Fourir ri to utio ( o iiit priod [] Thi ppr dvlop o o th udmtl topi i lyi d i PDE, mly orthogol pio Diitio: Th t o utio {Y (:,, } h o whih i piwi otiuou i iiit or iit itrvl [α,β], i id to orthogol i [α,β] with rpt to th wight utio r(>, i β Ym, Y r Ym Y d or ll m d α β r Y d or ll α W hll lwy um tht r( h oly iit umr o zro i [α,β] d th itgrl β r Y d,, it α Diitio: Th orm o utio Y ( i dotd y Y did th ir produt o utio with itl d writt β Y r Y d α A rl - vlud utio Y ( i lld qur - itgrl o th itrvl i [α,β] with rpt to th wight utio r( wh β α r Y d < + Th orthogol t {Y (:,, } i [α,β] with rpt to th wight utio r( i id to orthoorml t i r Y β d or ll [8] α I {Y (} i orthoorml t o utio th β m m ( α Y, Y r Y Y d δ m i m Whr δm i th Krokr dlt [6] i m Thu, orthoorml utio hv th m proprti orthogol utio, ut, i dditio, thy hv ormlizd [5], i, h utio Y ( o th orthogol t h dividd y th orm o tht utio, whih i β did Y r Y d H i α orthogol t o utio {Y (} i did o th itrvl [α,β], with Y w lwy otrut orthoorml t o utio X ( y diig Y X, α β [] Y I t i viw o (, β r Ym Y d X m,x ϕm, ϕ δm Ym Y Ym Y α d h X or ll Empl: Th t o utio Y i,or,,, i t o orthogol

2 5 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio utio ovr th itrvl <X< With rpt to th wight utio r( Thi i how y lultig th ir produt m m Ym, Y i,i i i d i ( m i ( m +, m, ( m ( m+ d Th orm qurd or h utio i giv y m m m m m Y, Y Y i d o d m i m For m,, Th rult r writt i ir produt ottio i m m i,i δm i m Th mir ppr diu priodi utio whih pdd i trm o iiit um o i d oi i whih mot utio outrd i girig r priodi utio Diitio: Fourir trigoomtri ri o utio o ( did o i did o - X, i th iiit trigoomtri ri ~ + o + i, Who oiit r giv y th ir produt ormul ( t dt t ( o dt, t ( i dt, Howvr, i th utio ( i odd, th i ( o tdt, ( t i tdt ( t i tdt, th Fourir trigoomtri ri rdu to th Fourir i ri: Whr o ~ i ( t i tdt, Thu, w olud tht i ( i odd, or did oly o(, d w mk it odd tio th th Fourir i ri otid Etly, i th m wy i ( i v, or did oly o (, d w mk it v tio th i ( t o tdt ( t o tdt, ( i tdt, th Fourir trigoomtri ri rdu to th Fourir oi ri: Whr ~ + o ( ( otdt, Empl: W hll id th Fourir oi ri o th < < Clrly, utio, to tdt tdt,, i t i t t o tdt t dt o (, + Thu, rom (, w hv ( ~ + o, < < Thorm: (Fourir thorm [ [7] ]t ( d ( piwi otiuou i th itrvl [-,] Th, th Fourir trigoomtri ri o ( ovrg to + _ ( + ( t h poit i th op itrvl (-, + _ ( + ( d t ± th ri ovrg to Empl: Coidr th utio [ [,,,, Clrly, CP (, d h igl jump diotiuity t For thi utio, th Fourir ( ( trigoomtri oiit r,, Thu, w hv ( ~ + i (, y F (3

3 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 6 From Fourir thorm i (,3 th qulity F( ( hold t h poit i th op itrvl (-, d (, whr t th right hd id i / whih i th m + _ ( + id i gi / whih i th m Alo, t ± th right hd + _ ( + Th Fourir itgrl i turl tio o Fourir trigoomtri ri i th tht it rprt piwi mooth utio who domi i mi-iiit or iiit [] A priodi utio ( did i iit itrvl (-, prd i Fourir ri y tdig thi opt, o priodi utio did i -<< (or ll prd Fourir itgrl t p ( priodi utio o priod p tht rprtd y Fourir ri ( i, + o + Whr dt o tdt, i dt, p Th prolm w hll oidr i wht hpp to th ov ri wh or thi w irt d, to oti o ( otdt ( dt + + i ( i tdt ( + W ow t + Th, d w my writ th Fourir ri i th orm ( dt ( o ( otdt ( + + (i ( dt Thi rprttio i vlid or y id p, ritrrily lrg, ut id W ow lt d um tht th rultig lim i olutly opriodi utio i itgrl o th -i, i, d < Th,, d th vlu o th irt trm o th right id o ( pproh zro Alo, d th iiit ri i ( om itgrl rom to, whih rprt (, i, o ( otdt (5 + ( tdt Now i w itrodu th ottio A ( otdt, B ( tdt (6 Th (5 writt [ + ] Ao B d (7 Thi rprttio o ( i lld Fourir itgrl Thorm: (Fourir Itgrl Thorm: t (, - << piwi otiuou o h iit itrvl, < (-, i, i olutly itgrl d d, o(-, Th, ( rprtd y Fourir itgrl (7 A o + B d Furthr, t h ( + + ( Empl: Fid th Fourir itgrl rprttio o th igl pul utio i < i > From (6 w hv A ( otdt otdt B t tdt Thu, (7 giv th rprttio o i d (8 Now rom thi Thorm it i lr tht i < o ( + d i ± (9 i >

4 7 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio Thi itgrl i lld Dirihlt diotiuity tor Sttig i (9 yild th importt itgrl d ( kow th Dirihlt itgrl Thorm : (Fourir Coi Itgrl Thorm: I ( tii th Dirihlt oditio o th o gtiv rl li d i olutly itgrll o (,, th, whr A A o tdt t otdt Thorm : (Fourir Si Itgrl Thorm: I ( tii th Dirihlt oditio o th o gtiv rl li d i olutly itgrll o (,, th ; whr B B i d t tdt Idd, i ( i v utio, th B( i (3 t otdt d th Fourir itgrl (7 d A rdu to th Fourir oi itgrl, A o td ( Similrly, i ( i odd, th i (6w hv A( t tdt d th Fourir itgrl (7 d B rdu to th Fourir i itgrl B i dtd ( i Empl: Epr Fourir i > i itgrl d h vlut o o ( Th Fourir i itgrl or ( d i i d ( tdt d ( tdt o d o d o i < d i > At, whih poit o diotiuity o (,th vlu o th ov itgrl ( + ( + + W ot tht (5 i th m ( [ oo t] dtd + ( o( d Th itgrl i rkt i v utio o, w dot it y F ( Si o( i v utio o, th utio do ot dpd o, d w itgrt with rpt to t (ot, th itgrl o F ( rom to i / tim th itgrl o F ( rom - to Thu, t t dt d From th ov rgumt it i lr tht ( o( dtd A omitio o (3 d ( giv o( (3 ( i ( t t dtd (5 Thi i lld th ompl Fourir itgrl From th ov rprttio o (, w hv t i ( dt d (6 Diitio: I th Fourir itgrl o ( i th ompl orm giv y t i ( dt d th prio i rkt i utio o, i dotd y F( or F( d i lld Fourir trorm o Now writig or t w gt F d (7 Ad with thi (6 om F d (8 Th rprttio (8 i lld th ivr Fourir trorm o F( Filly, i Thorm-, i (, - << i piwi otiuou o h iit itrvl,

5 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 8 Th, th Fourir trorm (7 d d < o ( it Furthr, t h, F d ( + + ( Empl: Fid th Fourir trorm o th qur wv utio From (7, w hv, <,, > F i d d i Furthr it ollow tht,,, <, i d, < < i,, < Th i thiqu or olvig prtil dirtil qutio (PDE o oudd ptil domi i th Fourir mthod [] Th mir ppr dl with th prolm o th Fourir trorm mthod or prtil dirtil qutio oidrig irt prolm i iiit domi whih tivly olvd y idig th Fourir trorm or th Fourir i or oi trorm o th ukow utio Howvr, or uh prolm uully th mthod o prtio o vril do ot work u th Fourir ri r ot dqut to yild omplt olutio Thi i du to th t tht ot th prolm rquir otiuou uprpoitio o prtd olutio I thi mir ppr w gi y motivtig th otrutio y ivtigtig how Fourir ri hv th lgth o th itrvl go to iiity Thror, thi ppr dvlop th thory o th Fourir trorm mthod or prtil dirtil qutio i whih qulittiv thory it Th mi ojtiv o thi ppr i to diu Fourir trorm mthod or prtil dirtil qutio (PDE whih ot hlp ull i pproimtio to th tru itutio d tht mor rliti modl would ilud om o th priodi utio writt i trm o iiit um o i d oi ri y uig Fourir trorm whih wr omplitd wh w r uig Fourir ri Trorm o Prtil Drivtiv Diitio: I th Fourir itgrl o ( i th ompl orm giv y t i ( dt d th prio i rkt i utio o, i dotd y F( or F( d i lld Fourir trorm o Now writig or t w gt F d ( Ad with thi (8 om F d ( Th rprttio ( i lld th ivr Fourir trorm o F( Filly, i Thorm-, i (, - << i piwi otiuou o h iit itrvl, d d < Th, th Fourir trorm ( o ( it Furthr, t h, F d ( + + ( Empl: Fid th Fourir trorm o th qur wv utio, <,, > From (, w hv F i d d i,, Furthr it ollow tht, <, i d, < < i,, < Thorm (Covolutio Thorm: Suppo tht ( d g( r piwi otiuou, oudd, d olutly itgrl utio o th -i Th F * g F( F( g (3

6 9 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio Whr *g i th ovolutio o utio d g did ( * ( ( g t g t dt g t t dt Proo: By th diitio d itrhg o th ordr o itgrtio, w hv F ( * g ( T g ( T dtd ( T g ( T ddt Now w mk th utitutio -Tv, o tht T+ν d ( T+ ν F ( * g ( T g ( v dvdt T F( Gg v T dt g v dv By tkig th ivr Fourir trorm o oth id o (3 d writig F( d F( g g otig tht d ( *, d l h othr, w oti g g d ( Th olutio o IBVP oitig o prtil dirtil qutio togthr with oudry d iitil oditio olvd y th Fourir trorm mthod I o dimiol oudry vlu prolm, th prtil dirtil qutio ily trormd ito ordiry dirtil qutio y pplyig uitl trorm Th rquird olutio i th otid y olvig thi qutio d ivrtig y m o th ompl ivrio ormul or y y othr mthod I two dimiol prolm, it i omtim rquird to pply th trorm twi d th dird olutio i otid y doul ivrio Suppo tht u(, i utio o two vril d t, whr -<< d t> Bu o th pr o two vril, r i dd i idtiyig th vril with rpt to whih th Fourir trorm i omputd For mpl, or id t, th utio u(, om utio o th ptil vril, d uh, w tk it Fourir trorm with rpt to th vril W dot thi trorm y u(, Thu, Fu ( ( t, u(, u( t, d Thi trorm i lld Fourir trorm i th vril [] To illutrt th u o thi ottio w omput om vry uul trorm Fourir Trorm d Prtil Drivtiv Giv u(, with -<< d t>, w hv d F( ( u( t, u(, ; dt d F( ( u( t, u(,,,, ; dt F( ( u( t, u(, ; F( ( u( t, ( u(,,,, To prov (i w trt with th right id d dirtit udr th itgrl ig with rpt to t: d d u (, u (, d dt dt u (, Th lt prio i th Fourir trorm o u( t, utio o, d (i ollow Rptd dirtitio udr th itgrl ig with rpt to t yild (ii Fourir Si d Coi Trorm For v utio ( Fourir itgrl i th Fourir oi itgrl (5 whr A( i giv y (6 W t A F, whr idit oi Th rplig t y W gt d d F o d (5 F od (6 Formul (5 giv rom ( w utio F ( lld th Fourir oi trorm o ( whr (6 F, d w ll it th ivr giv k ( rom Fourir oi trorm o F Rltio (5 d (6 togthr orm Fourir oi trorm pir Similrly, or odd utio ( th Fourir i trorm i d th ivr Fourir i trorm i F i d (7 d th ivr Fourir i trorm i F d (8

7 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 5 Empl : Fid th Fourir oi trorm o >, > Solutio F od od ( th itgrl i v th imgiry prt itgrt d to zro u it i odd ( y orm F diitio o Fourir tr y Empl : Coidr th utio i i i > Epr uig ivr Fourir oi d th ivr i trorm Solutio: W irt trt y omputig th Fourir oi trorm From (8, ( o o + F io d Uig (6, w oti th ivr oi trorm rprttio ( o o + od( W omput th i trorm imilrly y uig (6 F o o ( d thu th ivr i trorm rprttio ( i i i d( Coi d Si Trorm o Drivtiv o Futio I ( i olutly itgrl o th poitiv -i d piwi otiuou o vry iit itrvl, th th Fourir oi d i trorm o it Furthrmor, it i lr tht F d F r lir oprtor, i, F + g F ( + F ( d F + g F + F Thorm: t ( otiuou d olutly itgrl o th -i, lt ( piwi otiuou o h iit itrvl, d lt ( Th, ' F ( F ( F F [ ] ( (9 ' ( Proo: To how (i, w itgrt y prt, to oti ' ' FC ood o o + i d ( + F ( Alo w itgrt y prt, to oti ' ' FC d ( [ i i ( o o d ] F Similrly, ( ( '' ' ' FC F By ormul (ii with itd o giv ( ( '' ' ' FC ' ( F ( ( ' F ( '' ' ( F F ( ( By ormul ( with itd o giv '' ( ( ' F F h y (9 '' ( ( F F ( F ( ( W hv y th Fourir trorm, F d ( By imilr produr w id rltio tw th i d oi Fourir trorm o th drivtiv o utio, uh d FC ood d + d d d [ oo ] d( d ( itgrtig y pr α +

8 5 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio Udr th umptio, d α d Similrly, itgrtig, m d d d d (3 d d Equtio, ( d (3 yild, m α ( or '( or '( i odd d i tht w writ αo pl o '( rptig th produr my prd th um o d ithr W thu hv will our wh r r ( r + ( r Ad i α + r r + ( r + ( r α Similr produr with hlp o ( d (3 will yild α m r r r r d ( α + ( + r r + + r r ( α + + ( Similrly th ollowig rult r ily dduil, d (i o o d F d 3 df d F Wh, d 3 d d (ii ood F d d (iii d F d d F wh, F d d (iv d F d F Empl: W oud tht Applyig (ii with, w oti F F H F Covolutio thorm or Fourir i d oi trorm Thorm: t F ( d G trorm o ( d g(, rptivly, d lt F ( d G rptivly Th th Fourir oi th Fourir i trorm o ( d g( F [ F G ] g( ( + ( + d W hv F G od F odg ( od g( df ood g( d F [ o( o( ] d + + g( ( + ( + d Empl: (Covolutio with Coi Suppo tht i itgrl d v (- ( or ll g o Show tht, or ll rl umr ; d lt * g o( ( Solutio From th diitio d th t tht ; *gg*, w hv * g ( o [ ( ] dt ( o o( ( t o o ( dt + i i i Si i v, th produt ( it i odd, h, t it d o * g o( ( o o( dt ( o o( o o o( ( dt i i( o o o it ( ( dt (

9 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 5 3 Th Fourir Trorm mthod W ummriz th Fourir trorm mthod ollow: Stp : Fourir trorm th giv oudry vlu prolm i u(, d gt ordiry dirtil qutio i u u (, i th vril t Stp : olv th ordiry dirtil qutio d id (, Stp 3: ivr Fourir trorm u(, to gt u(, Thi mthod i uul i trtig vrity o prtil dirtil qutio, ut it h it limittio, i w hv to um tht th utio i th prolm d it olutio hv Fourir trorm Nvrthl, th mthod or u opportuiti yod th limittio, w ow illutrt Th mthod o olutio i t plid through th ollowig mpl Empl : W will how how th Fourir trorm ppli to th ht qutio W oidr th ht low prolm o iiitly log thi r iultd o it ltrl ur, whih i modld y th ollowig iitilvlu prolm u u < <, t>, > t u d u, iit, t > u,, < <, ( whr th utio i piwi mooth d olutly itgrl i (-, t u(, th Fourir trorm o u(, Thu, rom th Fourir trorm pir, w hv u(, u(, d u (, u (, d Aumig tht th drivtiv tk udr th itgrl, w gt u u u u (, u, t ( d (, d u t d I ordr or u(, to tiy th ht qutio, w mut hv u u u(, + u(, d Thu, u mut olutio o th ordiry dirtil qutio u + u Th iitil oditio i dtrmid y u (, u (, d d F Thror, w hv d h Now i (, u t F t t u( t, F d (5 /( t i d, t i i ( w dot F d g th rom (5 it ollow tht ( µ /( t u( t, ( µ dµ t (6 ( µ /( ( µ dµ t Thi ormul i du to Gu Thi ormul i du to Gu d Wirtr For h µ th utio ( µ /( u t, / t i olutio o th ht qutio d i lld th udmtl olutio Thu, (6 giv rprttio o th olutio otiuou uprpoitio o th udmtl olutio Th tdrd orml ditriutio utio Ф i did ( µ /( / t Thi i otiuou irig utio with Ф(, Ф(,Ф(, I <, th w writ ( µ t z t ( µ dµ dz, z t t t ( ( t z t z dz dz (7 Ф Ф t t

10 53 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio From (6 d (7 it i lr tht th olutio o th prolm C writt ut u, < <, t>, < u(,,, > u( t, Ф Ф t t Now uig th proprti Ф(, Ф(,Ф(, w vriy tht, <, lim u( t,, < < t,, > Empl : Coidr th prolm ut u, >, t>, u,, t> (8 u d u iit, t > u,, > whih ppr i ht low i mi iiit rgio I (8 th utio i piwi mooth d olutly itgrl i [, W di th odd utio Th rom (6 w hv ( µ /(, > (, < µ dµ t ( µ /( ( µ /( ( µ dµ + ( µ dµ t t I th irt itgrl w hg µ to -µ d u th odd o, to oti ( µ /( ( + µ /( µ dµ µ dµ t t Thu, th olutio o th prolm (8 writt ( µ /( ( + µ /( u( t, ( µ dµ t Th ov produr to id th olutio o (8 i lld th mthod o img I logou wy it how tht th olutio o th prolm t u u, >, t>, u,, t> (9 u d u iit, t > writt u,, > ( µ /( ( + µ /( u( t, ( µ dµ t Hr, o our, w d to td ( to v utio, > (, < I (9 th phyil igii o th oditio u, i tht thr i prt iultio, i, thr i o ht lu ro th ur Empl 3: Coidr th iitil-vlu prolm or th wv qutio ut u, < <, t>, > u d u iit, t > ( u,, < <, u,, < <, t whr th utio d r piwi mooth d olutly itgrl i (-, To id th olutio o thi prolm, w itrodu th Fourir trorm Fj j d, j, d it ivrio ormul j Fj d, j, W lo d th Fourir rprttio o th olutio u(,, Whr u(, u( t, u (, d i ukow utio, whih w will ow dtrmi For thi, w utitut thi ito th dirtil qutio (, to oti u(, u(, t + d Thu, u mut olutio o th ordiry dirtil qutio

11 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 5 u + u who olutio writt (, + u t ot t To id ( d, w ot tht u(, d u (, d d h F d F Thror, it ollow tht F u (, F ot + t d h th Fourir rprttio o th olutio i F u (, F ot + t d i i i i + Now i o, i w hv i F ( o d F ( d + ( + t ( F ( d + ( + t + ( t Similrly, ( ξ + t ξ F d dξ t + t t t F d t t F d ( + ( F + t ξ t dξ F dξ d d Puttig th togthr yild d Almrt ormul + t t u (, [ ( + t + ( t ] + ( ξ dξ Empl : Coidr th ollowig prolm ivolvig th pl qutio i hl-pl: u + uyy, < <, y > u(,, < < uy (, M, < <, y>, whr th utio i piwi mooth d olutly itgrl i (-, I, th w lo hv th implid oudry oditio lim uy,, lim uy, For thi, w lt, y +, F d, F d d u(,y u (, y d W id tht u + u u, y + d u, yy Thu, u (, y y mut tiy th ordiry dirtil qutio u y u d th iitil oditio u(, F or h Th grl olutio o th ordiry dirtil y y qutio i + I w impo th iitil oditio d th oudd oditio, th olutio om y F, u(, y F y F, < y Thu, th dird Fourir rprttio o th olutio i y u(,y F d To oti pliit rprttio, w irt th ormul or F( d ormlly itrhg th ordr o itgrtio, to oti

12 55 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio ξ y u(,y ( ξ dξ d Now th ir itgrl i ( ξ y d ( ξ dξ ( ξ y ( ξ y d R d y R yi ( ξ y + ( ξ Thror, th olutio u(,y pliitly writt y ξ ( y + ( ξ u(,y ( ξ d Thi rprttio i kow Poio itgrl ormul I prtiulr, or, < < u(,, othrwi Thu, ( om y dξ / y y + ( ξ ( ξ + y (,y dξ u ξ Uig th utitutio v w hv dξ ydv, o y tht (,y u ( / y + v ( / y t t y y Th Fourir Si d Coi Trorm Mthod W will motivt th itrodutio o Fourir i d oi trorm y oidrig impl phyil prolm I ordr to u th Fourir i d oi trorm to olv prtil dirtil qutio: I th oudry oditio r o th Dirihlt typ: whr th utio vlu i prrid o th oudry, th th Fourir i trorm i ud I th oudry oditio r o th Num typ: whr th drivtiv o utio i prrid o oudry, th Fourir oi trorm i pplid I ithr, th PDE rdu to ODE i Fourir trorm whih i olvd Th th ivr Fourir i (or oi trorm will giv th olutio to th prolm Iiit Fourir oi d i Trorm Mthod To olv prtil dirtil qutio (otiig od drivtiv did o mi-iiit itrvl, dv uig Fourir oi trorm, ( mut kow I ut (,, i giv th w mploy oi o u trorm to rmov Diitio: Th iiit Fourir oi trorm o utio ( or <<,i did F i i d ig poitiv itgr Hr ( i lld th ivr Fourir oi trorm o F ( d i did F ( o d Similrly, th Fourir i trorm my ud or mi-iiit prolm i ( i giv Furthrmor, prolm r mor rdily olvd i th oudry oditio r homogou Thu, i (, prtio o vril motivt th u o i oly Similrly, ( impli th u o oi Diitio: Th iiit Fourir i trorm o utio ( o uh tht << i dotd y F (, ig poitiv itgr d i did F i i d Hr ( i lld th ivr Fourir i trorm o F ( d did F ( i d Empl: W hll mploy th Fourir i trorm to id th olutio o th ollowig prolm ivolvig th pl qutio i mi-iiit trip: u + uyy, < <,< y < u(,, < <, u(, y, < y < u,, < <, whr th utio i piwi mooth d olutly itgrl i [, W hll lo d th oudry lim uy, lim u y, oditio d d, For thi, w lt, F d, F d u (,y u (, y d

13 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio 56 W id tht u (, y u, + uyy u (, y + d y Thu, u d h mut tiy th ordiry dirtil qutio u y u + u, y oh y ih y Now th oudry oditio u ( Thu, w hv, yild ih oh ih u (, y oh y + ih y oh ih ( y oh Now i u (, F F ih, d thror ih (, u y F Thi giv th olutio ih ( y ih, w id ih ( y u (,y F d ih ih ( y ( t t dtd ih Fiit Fourir oi d i Trorm Mthod Wh th domi o th phyil prolm i iit, it i grlly ot ovit to u th trorm with iiit rg o itgrtio I my, iit Fourir trorm ud with dvtg Diitio: Th iit Fourir i trorm o,< < i did F ( i i d Whr i itgr Th utio ( i th lld th ivr iit Fourir i trorm o F ( d i giv y F( i F( i ( Diitio: Th iit Fourir oi trorm o,< < i did F ( o d, Whr i itgr Th utio ( i th lld th ivr iit Fourir oi trorm o F ( d i giv y F + F( o( o Fiit Fourir trorm r uul i olvig prtil dirtil qutio For thi, w ot tht (, u t i d u( t, i u( t, o d Ad h u F F( d imilrly, u F F( [ u(, u( t, o ], u u F F (3 F (, (, o + u t u t u F F (, (, o u t u t Empl: Fid th olutio o th prolm u u, < <, t > u,,< < u, t u, t, t > Tkig th iit Fourir i trorm with o oth id o th prtil dirtil qutio giv d u u i i d Writig u or F ( d uig (3 with u, t, u, t ld to du(, t dt 6, whih olvd to oti u (, u /6 Now tkig th iit Fourir i trorm o th oditio u( w hv

14 57 Itrtiol Jourl o Prtil Dirtil Equtio d Applitio u (, i d o / i / / /6 3 o Si u (, it ollow tht 3 u( t, o Thu, rom ( w gt t /6 6 o t (, /6 u t i 3 Coluio Howvr, Phyil prolm vr I thi work, wh modlig prolm ovr rgio tht tdd vry r i t lt o dirtio, w ot idlizd th itutio to tht o prolm hvig iiit tt i o or mor dirtio, whr y oudry oditio tht would hv pplid o th r-wy oudri r dirdd i vor o impl oudd oditio o th olutio th pproprit vril i t to iiity Suh prolm wr mthmtilly modld y dirtil qutio did o iiit rgio For o-dimiol prolm w ditiguih two typ o iiit rgio: iiit itrvl tdig rom - to d mi-iiit itrvl tdig rom o poit (uully th origi to iiit (uully + r iiit, ut y itroduig mthmtil modl with iiit tt, w r l to dtrmi hvior o prolm i th itutio i whih th ilu o tul oudri r ptd to gligil Thu th mir ppr dvlopd th Fourir trorm mthod d pplid it to olv: ht low prolm o iiitly log thi r iultd o it ltrl ur, ht low i mi iiit rgio, wv qutio, pl qutio i hl-pl d i miiiit trip, d om prtil dirtil qutio o th tir rl li Ev though urvy o thi mir ppr how tht wht i tully tudid Fourir trorm mthod to PDE i tht, w tk th Fourir trorm o PDE d it iitil d oudry oditio to rdu it ito ODE W th olvd thi ODE or th trormd utio W ivrtd thi utio to dtrmi th olutio to our PDE Thi i ot jut mthod tht i pii to th Fourir trorm u thi mthod lo work or th pl trorm d i grl or my itgrl trorm Th itgrl diig th Fourir trorm d it ivr r rmrkly lik, d thi ymmtry w ot ploitd, or mpl wh mlig ppdi giv or Fourir trorm O oditio o thi i tht th vril you tk to th itgrl trorm it domi mut mth th rg o itgrtio o th itgrl trorm Th typ o oudry d iitil oditio tht r giv hould lo plyd rol i whih trorm hould ud I, th Fourir trorm i ud to lyz oudry vlu prolm o th tir li Th tio o Fourir mthod to th tir rl li ld turlly to th Fourir trorm, trmly powrul mthmtil tool or th lyi o opriodi utio It i rol to pt Fourir trorm mthod pply to olv dirt orm o prtil dirtil qutio uh Tlgrph qutio: utt + ( α + β ut + αβu u or th αβ>, Th Fourir trorm i o udmtl import i rod rg o pplitio, iludig oth ordiry d prtil dirtil qutio, qutum mhi, igl d img proig, otrol thory, d proility, to m ut w Rr [] Agrwl R d R Dol, 9 Ordiry d prtil dirtil Equtio With Spil Futio, Fourir Sri, d Boudry vlu Prolm, Sprig Strt, Nw York [] Amr N, Prtil Dirtil Equtio With Fourir Sri Ad Boudry Vlu Prolm, d ditio, Plo Prti Hll, Uitd Stt o Amri [3] Bdru V, 8 Highr Egirrig Mthmti, Tt M Grw-Hill Pulihig Compy imitd, Nwdlhi [] Duy D, 998 Advd Egirig Mthmti, CRC Pr C, Nw York [5] Hrm R, 987 Elmtry Applid prtil Dirtil qutio With Fourir ri d Boudry Vlu Prolm, Eglwood Cli, Nw Jry [6] Piku A d Zry S, 997 Fourir Sri d Itgrl Trorm, Cridg Uivrity Pr, Uitd Kigdom [7] Puri P, 997 Prtil Dirtil Equtio d Mthmti, CRC Pr, Nw York [8] Tvito A, 998 Itrodutio to Prtil Dirtil Equtio A Computtiol pproh, Sprigr-Vrlg, Nw York, I