CONTENTS CHAPTER 1 POWER SERIES SOLUTIONS INTRODUCTION POWER SERIES SOLUTIONS REGULAR SINGULAR POINTS 05

Transcription

1 CONTENTS CHAPTER POWER SERIES SOLUTIONS 3 INTRODUCTION 3 POWER SERIES SOLUTIONS 4 3 REGULAR SINGULAR POINTS 5 FROBENIUS SERIES SOLUTIONS 4 GAUSS S HYPER GEOMETRIC EQUATION 7 5 THE POINT AT INFINITY 9 CHAPTER SPECIAL FUNCTIONS LEGENDRE POLYNOMIALS BESSEL FUNCTIONS GAMMA FUNCTION 5 CHEPTER 3 SYSTEMS OF FIRST ORDER EQUATIONS 3 LINEAR SYSTEMS 3 HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS 33 NON LINEAR SYSTEM 4 CHAPTER 4 NON LIEAR EQUATIONS 6 4 AUTONOMOUS SYSTEM 6 4 CRITICAL POINTS & STABILITY 8 43 LIAPUNOV S DIRECT METHOD 3 44 SIMPLE CRITICAL POINTS -NON LINEAR SYSTEM 34 CHAPTER 5 FUNDAMENTAL THEOREMS 38 5 THE METHOD OF SUCCESSIVE APPROXIMATIONS 38 5 PICARD S THEOREM 39 Diereial Eqaios 3

2 CHAPTER 6 FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 46 6 INTRODUCTION REVIEW 46 6 FORMATION OF FIRST ORDER PDE CLASSIFICATION OF INTEGRALS 5 64 LINEAR EQUATIONS PFAFFIAN DIFFERENTIAL EQUATIONS CHARPIT S METHOD 6 67 JACOBI S METHOD CAUCHY PROBLEM 7 69 GEOMETRY OF SOLUTIONS 74 CHAPTER 7 SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS 7 CLASSIFICATION 78 7 ONE DIMENSIONAL WAVE EQUATION 8 73 RIEMANN S METHOD LAPLACE EQUATION HEAT CONDUCTION PROBLEM Diereial Eqaios 4

3 Irodcio CHAPTER POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS A algebraic cio is a olomial a raioal cio or a cio ha saisies a olomial eqaio whose coeicies are olomials The elemear cios cosiss o algebraic cios he elemear rascedeal cios or o algebraic cios- he rigoomeric cios ad heir iverses eoeial ad logarihmic cios ad all ohers ha ca be cosrced rom hese b addig or mlilig or akig comosiios A oher cio is called a secial cio Cosider he ower series a The series has a radis o covergece R R sch ha he series coverges or < R ad diverges or > R We have R = lim a a For he geomeric series R = ad or he eoeial series R =! ad he series! coverges ol or = Sose a = or < R The has derivaives o all orders ad he series ca be diereiaed erm b erm ' a '' a ad so o ad each series coverges or < R I ac we ge a! A cio which ca be eaded as a ower series a valid i some eighborhood o is said o be aalic a Polomials e si cos are aalic a all ois b / + is o a = - Diereial Eqaios 5

4 Power series solios I ma be recalled ha ma diereial eqaios ca o be solved b he ew aalical mehods develoed ad hese mehods ca be emloed ol i he diereial eqaios are o a ariclar e B alig he ollowig mehod solios ca be obaied as a ower series ad hece kow as ower series mehod ' Cosider he eqaio We ma assme ha his eqaio has a ower series solio i he orm = coverges or < R or some R a ha ' The a a 3a Sice 3 ' b eqaig he coeicies o like owers o we ge a=a a=a3a3=a which redces o a=aa=a/ = a/!a3=a/3! Ths we obai = a + /! + /!+ = a e where a is le deermied ad hece arbirar Now le s cosider he geeral secod order homogeeos eqaio '' ' P Q * I boh P ad Q are aalic a = we sa is a ordiar oi o he eqaio We ma assme he solio o he eqaio * as a ower series = a valid or - < R or some R The varios coeicies ca be od i erms o a ad a which is le deermied Assme = '' Cosider Here P= ad Q = which are aalic a = a The he eqaio gives he recrrece relaio ++ a++a= or = Sbsiig =sccessivel ad redcig we ge a+ = - a/ ! ad a=- a/! Hece = a a! 4! 3! 5! = a cos + a si cosa '' ' Cosider he Legedre s eqaio Here P = ad Q = which are aalic a = where is a Diereial Eqaios 6

5 Le = a The he eqaio gives he recrrece relaio ++a+--a- a++ a= P = which gives a a a3 a! 3! a 3 a 4! a 3 4 5! 4 5 a Ths = 3 4 a! 4! a 3! 5! The radis o covergece or each o he series i he brackes is R = The series i he irs bracke ermiaes or = 46 ad he series i he secod bracke ermiaes or = 35 The reslig olomials are called Legedre olomials whose roeries will be discssed laer '' E The eqaio 4 where is a cosa has a ower series solio = a a = Show ha he coeicies are relaed b he hree erm recrrece relaio a a 4a I he deede variable 4 is relaced b = w e '' ' show ha he eqaio is rasormed o w w w ad is ower series solio a = ivolves ol a wo erm recrrece relaio 3 Reglar siglar ois = is a siglar oi o * i eiher P or Q is o aalic a I his case he ower series solio ma o eis i a eighborhood o B he solios ear a siglar oi is imora i a hsical coe ad mos o he cases he eis Origi is a siglar '' ' oi o ad or > = c + c - is is geeral solio A siglar oi o * is called reglar siglar i boh -P & - Q are aalic a or which = '' ' Cosider he Legedre s eqaio - are siglar ois b he are reglar siglar For he Bessel eqaio o order Diereial Eqaios 7

6 '' ' where is a o egaive cosa = is a reglar siglar oi I = is reglar siglar oi o * he b deiiio -P & - Q are aalic a ad hece we ma ake -P = ad - Q = = eeced m q a A solio o he eqaio * as a Frobeis series where m is a real mber ad a is assmed o ero ca be O sbsiig = m a i * ad eqaig he coeicies we ge he recrsio ormla a [ m m m q ] a [ m k q ] k k k k lim lim ** Here P ad q Q For = ** gives m m m q *** called he idicial eqaio which deermies he vales o m Sbsiig he vales o m ad akig =3 i ** a s ca be deermied i erms o a ad he resecive solios ca be obaied '' ' Eg Cosider he eqaio = is a reglar siglar oi o he eqaio Le s assme ha he solio a = is = we ge he idicial eqaio m m m -- m = -/ For m = -/ resecivel we ge he solios o deermiig he a s sccessivel rom he recrrece 4 3 relaio ** as a ad a which 5 35 are ideede also ad hereb he geeral solio is = c + c where c& c are arbirar cosas m a Diereial Eqaios 8

7 Remark Le he roos o he idicial eqaio be real sa m& m wih m m The he eqaio * has a Frobeis series solio corresodig o m he larger eoe I m = m here is o scoe o ge a secod ideede solio b he same rocedre ad i ma be od b some alerae mehod I m - m is o a osiive ieger aoher ideede solio corresodig o m ca be obaied ad oherwise he mehod ma o be givig a secod ideede solio E Cosider he eqaio '' 3 ' 4 4 Show ha = is a reglar siglar oi ad id he ol oe Frobeis series solio E Fid he wo ideede solios o '' ' a = E3 The Bessel eqaio o order = ½ amel '' ' has = as a 4 reglar siglar oi The eoes m & m is sch ha m m = b he mehod gives wo ideede solios ad deermie hem 4 GAUSS S HYPER GEOMETRIC EQUATION '' ' The eqaio [ c a b ] ab where a b c are cosas A rereses ma classical eqaios ad is kow as Gass s Her geomeric eqaio We have P = c a b ad Q = ab The ol siglar ois are = & ad he are reglar siglar ois We ma roceed o ivesigae he solio a = We ge = c & q = so he idicial eqaio is mm-+mc = which gives m = & m = -c I -c is o a osiive ieger ie i c is o ero or a egaive ieger he A has a solio o he orm = a coeicies o we ge he recrsio ormla ; Sbsiig i A ad eqaig o ero he a a b a Wih a= we ge c a a a b b b i sccessio all a s ad he solio =! c c c called he her geomeric cio deoed b Fa b c Sice R = lim a a = lim a b c = he series coverges or < Noe ha he series redces o a olomial or a or b eqal o ero or some egaive ieger Diereial Eqaios 9

8 I -c is o ero or a egaive ieger a secod ideede solio ca be obaied similarl or b he sbsiio = -c A becomes '' ' { c [ a c b c ] ] } a c b c --B a her geomeric eqaio wih a b c relaced b a-c+ b-c+ ad -c Hece he solio o B a = is = Fa-c+b-c+-c or = -c Fa-c+b-c+-c whe c is o a osiive ieger Ths i c is o a ieger he he geeral solio o A a = is = c Fabc + c -c Fa-c+b-c+-c To id he solio a = we ma ake = - so ha whe = = '' ' A becomes [ a b c a b ] ab Hece he geeral solio a = whe c-a-b is o a ieger is = c Faba+b-c+- + c - c-a-b Fc-bc-ac-a-b+- Remark The solio o he geeral her geomeric eqaio A B '' C D ' H Where A B is obaied hrogh he ma = B A A which rasorms he eqaio o ' [ F G] H ad = A & = B o = & = resecivel E Show ha + = F- b b - log+ = F - si F 3 lim lim E Show ha e F a b a cos F a a b b a 4a '' ' E3 Cosider he Chebchev s eqaio where is a o egaive cosa Trasorm i io a her geomeric eqaio b = geeral solio ear = is = ad show ha is c F c F 3 ' ab E4 Show ha F a b c F a b c c Diereial Eqaios

9 E5 Show ha he ol solios o he Chebchev s eqaio whose derivaives are boded ear = are = cf 5 The oi a iii I is o racical imorace o sd he solios o a give diereial eqaio or large vales o B he rasormaio = / ad akig small his ca be achieved Cosider he Eler eqaio '' 4 ' which is rasormed o rasormed eqaio so is = b he sbsiio = / Sice = is a reglar siglar oi o he or he origial eqaio Cosider he her geomeric eqaio A B he ma =/ i is rasormed o a b c ab eoes m = a b Hece = Cole her geomeric eqaio = is a reglar siglar oi wih is also a reglar siglar oi wih eoes a b d d Cosider he her geomeric eqaio s s [ c a b s] ab ds ds '' a ' Chagig s o = bs he eqaio becomes c a b b which has he reglar siglar ois = b ad I we le b he b will be merged wih ad his colece o wo reglar siglar ois rodce a irreglar siglar oi a eqaio '' ' or he limiig eqaio c a called he cole her geomeric Diereial Eqaios

10 Legedre Polomials For a o egaive ieger CHAPTER SPECIAL FUNCTIONS LEGENDRE POLYNOMIALS '' ' cosider he Legedre s eqaio --L We are ow roceedig o id he solios o L boded ear = a reglar siglar oi Take = The = corresods o = ad he rasormed eqaio is d d [ ] her geomeric eqaio = is reglar siglar wih idicial eqaio mm- + m = givig he ol eoe m = The corresodig Frobeis series solio is = F - + Le a secod ideede solio be =v where v ' e e a a sice is a olomial wih o ero cosa erm Ths v = log + a + ad = log + a + As log is boded a = ie a = Ths he ol boded solios o L boded a = are cosa mliles o = F - a olomial o degree called he h Legedre olomial deoed b P We ma roceed o eress he olomial P i he sadard ower orm ad obai a geeraig ormla kow as he Rodrige s ormla The ower series solio we have obaied earlier a = redces o a olomial o degree sice = a o egaive ieger ad here b a valid solio boded a = also Ths b he above observaio abo boded solios a = we ge he earlier solio as a cosa mlile o P O simliicaio P= --!!!! B P is olomial o degree which coais ol odd or eve owers o accordig as is odd or eve Hece P=a +a Diereial Eqaios

11 I is oed rom ha P= ad sig P-=- Frher rom we ge a=!! Sice a olomial solio is valid everwhere rom he ower series solio we have obaied a = he recrsio ormla sed i ha coe relaes he coeicies o k k P i he orm Ths a k ak ad wriig i he reverse order k k wih k = - -4 ields a a 3 3 a 4 a = a Ths P =!! The coeicie o -k i 3 ca be simliied as k k! ad we obai k! k! k! P = k k k! -k = k! k! k! k k k d k! k! d d! d k k =! d k k! k!! d which is sed or comig he Legedre Polomials direcl We ge P= P= P=/3 - P3=/5 3-3 E Assmig ha 3 P+= ad P=! P called he Rodrige s ormla is re show ha P= P-=- B diereiaig boh sides wro ad eqaig he coeicies o obai he recrsio ormla + P+ P P ad se i o id P & P3 rom P= ad P= Orhogoali o Legedre Polomials i m Pm P d i m ie {P = } is a amil o orhogoal cios i [-] Le be a cio wih a leas coios derivaives i [-] ad cosider he iegral I= P d = d d! b Rodrige s ormla d Diereial Eqaios 3

12 Alig iegraio b ars I = d! d - d! d d d = - d! sice he eressio i bracke vaishes a boh he limis d Coiig o iegrae b ars we ge I = d! d d = d! Take = Pm where m < The = sice Pm is a olomial o degree m Ths I = P P d m Now le = P The = m!! ad we ge rom above I =!! d! d! sbsiio = si!! cos d b he! Ths I =! = 3 o simliicaio Legedre series Le be a arbirar cio he ap where a P d is called he Legedre series easio o The eressio o a s are moivaed b he orhogoali roeries o Legedre olomials Noice ha i P is a olomial o degree k he P = a P k Leas sqare aroimaio Le be a cio deied i [-] ad cosider he roblem o idig a olomial P o degree less ha or eqal o or a give sch ha he error esimae I = [ P ] d is leas We will show ha he aroimaio is iqel ied as Diereial Eqaios 4

13 P = ak P k where ak k Pk d ad Pk is he k h Legedre k olomial We have I = [ bk Pk ] d = [ ] d + k k k b k - k b k Pk d = [ ] d + k k b k - k k a k b k = [ ] d + b k a k k k Hece he resl - k k a k which is leas whe b k = ak or k = o E I P is a olomial o degree > sch ha P d or k = - show ha P = c P or some cosa c k! E Show ha amog all he moic olomials P o degree P! is he iqe oe so ha [ P ] d is leas Bessel cios The Gamma cio '' ' The diereial eqaio where is a o egaive cosa is kow as he Bessel diereial eqaio Noe ha = is a reglar siglar oi o he eqaio wih idicial eqaio m - = ad eoes are m = ad m = - The eqaio has a solio i he orm = a where a The recrrece relaio or a s is +a + a-= Sice a-= a = or odd vales o We ge Hece we have a! a = a! Takig a =/! we ge he solio Diereial Eqaios 5

14 J =! =! /! P called he! Bessel cio o he irs kid o order Remark: I he above discssio we have sed he oaio! hogh is a real mber o ecessaril a o egaive ieger or which acorials are deied We eed he deiiio o acorial wih he hel o gamma cio as ollows For > we have e The amos recrrece relaio o gamma iegral is obaied below Now e lim b b e b lim b = e e b b lim = e = b b sice as b b e Now e Ths or a o egaive ieger = =- = = -- =! From he recrrece relaio reseed as = -- I we ca deie or -<< sice is available or +> For -<<- we agai se I ad he eeded deiiio sice -<+< This rocess is coied o deie or all egaive real mbers which are o iegers Agai rom I we ge = lim = lim = ad sig I reeaedl deie = Hece we ca deie or a egaive ieger Now we have eeded or all vales o We ma ow deie!= or all vales o or is recirocal /! = / which vaishes b deiiio a a egaive ieger aloe Now wih he above eesio or acorial or is recirocal J is well deied or all > We have m - m = There eiss a Frobeis series solio corresodig o m = - / eve whe =/ 3/as a mlile o J- =!! Diereial Eqaios 6

15 The irs erm o his series is! which is boded as Hece J- is boded a = where as J is boded a = or o a ieger Ths or o a ieger he geeral solio a = is = c J + c J- For = m a o egaive ieger J m = sice or = m- m! m m / /! m!! m! m Ths J m = m m m / m!! - m m / - m Jm! m! Hece Jm & J-m are o ideede whe m = J Remark: The geeral solio is = c J + c Y where Y= lim or o a ieger ad or m = Ym = Y cos J si E Show ha / We have / e e / 4 d 4 e d e s ds b he sbsiio = s e dd 4 e r rddr Chagig o olar coordiaes Hece / E Whe = ½ show ha he geeral solio ca be ake i eqivale orms = c J/ + c Y-/ ad = d cos d si Hece J = a cos + b si ad J = c cos + d si Evalae abcd ad show ha J si ad J cos Proeries o Bessel cios We have J= d Now J d P /!! d d P =!! P =!! Diereial Eqaios 7

16 /! d d! ie J J = J- d d Similarl i ca be show ha J J ie J + - J = J- & - J J = - - J+ Now / J + / J =J- 3 & /- J / J=-J J = J--J+ 5 & 3-4 / J = J-+J+ 6 The recrrece relaio o Bessel cios is / J = J-+J+ Orhogoali roeries I s are he osiive eroes o J he J J m d J m m '' ' Le = J The I a & b are disic osiive cosas '' ' he = Ja & v = Jb sais he eqaios a -- '' ' ad v v b v -- v d d ' ' ' ' v v v v b a v --3 d ' ' 3 [ v v ] b a v --4 d Iegraig rom o we ge ' ' b a vd v v eroes o J Le a = m & b = The we have obaied J = i a & b are disic J d i m m Diereial Eqaios 8

17 ' '' ' ' ' a or d [ d ' a ] a --5 Ths [ ' a ] a d 6 B = Ja ad hece =a J a Ths we ge rom 6 wih a relaced b ha J d J ' J Bessel series Le be a cio deied i [] ad s be he osiive eroes o some ied Bessel cio J The a J where a = J J d is called he Bessel series easio o The ollowig heorem gives sicie codiios or he easio o a cio as a Bessel series Bessel Easio Theorem : Assme ha ad have a mos a iie mber o jm discoiiies i [] I < < he he Bessel series B coverges o whe is a oi o coii ad coverges o ½ [-++] whe is a oi o discoii E Prove ha he osiive eroes o J ad J+ occr alerael E I / J / / / show ha J where J / s are he osiive eroes o J E3 I F = i [ show ha is Bessel series or a give is J J I g is a well-behaved cio i [] he show ha g d J g J d B akig g = + dedce ha 4 ad 4 6 Takig = ½ derive ha 6 ad Diereial Eqaios 9

18 3 LINEAR SYSTEMS CHAPTER 3 SYSTEMS OF FIRST ORDER EQUATIONS Le be variables deedig o he ideede variable Cosider he ollowig ssem o irs order diereial eqaios d F d G The above ssem is called liear i he deede variables & are aearig ol i irs degree Ths he corresodig liear ssem ca be reseed as d a b d a b I & are ideicall ero he ssem is called homogeeos Ths he associaed homogeeos liear ssem is d a b d a b 3 We assme ha ai bi i i = are coios i some ierval [ a b ] The solio o is a air o cios = ad = We reqire he sor o he ollowig heorems i or discssio Theorem I is a oi i [ a b ] ad i ad are give mbers he has a iqe solio = = valid i [ a b ] sch ha = ad = Proo is give laer Theorem I he liear homogeeos ssem 3 has wo solios = = ad = = valid i [ a b he = c + c = c + c is also a solio or a wo cosas c c Le W = The W is called he Wroskia o he solios ad Diereial Eqaios

19 Theorem 3 I he wo solios ad o he homogeeos ssem 3 has a Wroskia ha does o vaish o [ a b ] he = c + c = c + c where c & c are arbirar cosas is he geeral solio o 3 i [ a b ] Theorem 4 The Wroskia o wo solios o he homogeeos ssem is eiher ideicall ero or o where ero i [ a b ] Proo: We have dw [ a + b ]W which gives W = cosa c The W i c = ad W or a i c a b ce or some Remark: The wo solios = = ad = = valid i [ a b ] o he homogeeos ssem are said o be liearl ideede i oe is o a cosa mlile o he oher which is eqivale o he codiio ha he Wroskia o he solios is o ero The ollowig Theorem is a coseqece o he above deiiio ad Theorem 4 Theorem 5 I he wo solios = = ad = = are liearl ideede he = c + c = c + c where c & c are arbirar cosas is he geeral solio o 3 i [ a b ] Theorem 6 I he wo solios ad o he homogeeos ssem are liearl ideede ad = = is a ariclar solio o he corresodig o- homogeeos ssem he = c + c + = c + c + where c & c are arbirar cosas is he geeral solio o i [ a b ] Proo: Le be a solio o The i ca be easil show ha - - is a solio o 3 ad he resl ollows b vire o Theorem 5 3 Homogeeos Liear Ssem wih cosa coeicies d a b Cosider he ssem 4 where a abb are cosas We ma d a b assme ha a solio o he ssem ca be ake as I we sbsie 5 i eqaio 4 Ae Be m m 5 we ge Ame m = a A e m + b B e m bme m = aa e m + b B e m Cacellig e m hrogho gives he homogeeos liear algebraic ssem a-ma+bb= aa+b-mb= 6 Diereial Eqaios

20 I is clear ha he ssem rivial solio A = B = ields he rivial solio a m b = = o 4 The ssem 6 has a o rivial solio i a b m O easio o he deermia we ge he qadraic eqaio m a + b m + ab ab = 7 wih roos sa m = m m For m = m he ssem 6 gives a o rivial solio sa A B The We ge he solio corresodig o m = m i a similar ashio as Ae B e A e B e The are o he roos m & m are imora wheever we r o wrie he geeral solio Case : Disic Real roos I m ad m are real ad disic he = c + c = c + c is he geeral solio Case Comle roos Le m = a ib be he roos o 7 For m = a + ib solve 6 o ge A = A +ia B = B+iB Sice we reqire real solios aloe he geeral solio is a liear combiaio o a a e A cos b A si b e B cos b B si b ad a a e A si b A cos b e B si b B cos b These are obaied b Ae searaig io real ad imagiar ars he solio Be Case 3: Two eqal real roos m m m m m m obaied or m = a +ib We ge oe solio as Ae Be m m A secod solio ma be obaied i he orm A A e B B e m m Eg Cosider he ssem ad heir liear combiaio gives he geeral solio d d 4 Le = A e m = B e m The aer cacellaio o e m we ge he liear algebraic ssem - m A + B = 4 A + - m B = For o rivial solio o he algebraic ssem we have m + m 6 = ie m = -3 or Wih m = -3 he algebraic ssem becomes 4 A + B = Diereial Eqaios

21 A o rivial is chose as A = B = -4 Ths we have he solio = e -3 = -4 e -3 Wih m = we ge - A + B = A o rivial solio is ake as A = B = This gives he solio = e = e I ma be oed ha he solios obaied are ideede Hece he geeral solio is = c e -3 + c e = -4 c e -3 + c e Eg Cosider he ssem d 3 4 d Le = A e m = B e m The aer cacellaio o e m we ge he liear algebraic ssem 3 m A 4 B = A + - m B = For a o ero solio 3 m - m + 4 = ie m m + = or m = Wih m = gives A B = Choose A = B = Corresodig solio is = e = e A secod solio liearl ideede rom he above is assmed o be = A + A e ad = B + B e The we obai A + A + A = 3 A + A 4 B + B & B + B + B = A + A B + B Sice hese are ideiies i we ge A 4 B = A B = A A -4 B = A B B = A o ero solio is ake as A = B = A = B = Now we ge aoher solio = + e = e The wo solios obaied are liearl ideede Hece we ge = c e + c + e = c e + c e as he geeral solio Eg3 Cosider he ssem d 4 d 5 Le = A e m = B e m The aer cacellaio o e m we ge he liear algebraic ssem m - 4 A + B = 5A + m B = For o rivial solio o we have m 6 m + 8 = or m = m are comle we are eecig comle vales or A & B also Le A = A + i A B = B + i B ad sbsie m = 3 3i i We obai 3 3i Sice he vales o - +3 i A + i A + B + i B = 5 A + i A i B + i B = Eqaig he real ad imagiar ars - A 3 A + B = 3 A A + B = 5 A B + 3 B = 5 A 3 B B = Cosider he coeicie mari ad redce i o row echelo orm A solio o he homogeeos algebraic ssem is A = A = B = B = -3 The geeral solio is = e 3 c cos 3 + d si 3 = e 3 [c cos si 3 + dsi 3 3 cos 3] Diereial Eqaios 3

22 33 No liear ssem Volerra s re redaor eqaios Cosider a islad ihabied b oes ad rabbis The oes h he rabbis ad rabbis eed o carros We assme ha here is abda sl o carros As he rabbis become large i mber oes lorish as he h o rabbis ad heir olaio grows As he oes become meros ad ea oo ma rabbis he rabbi olaio declies As a resl he oes eer a eriod o amie ad heir olaio declies As oes decrease i mber he rabbis become more sae reslig i a olaio srge As ime goes o we ca observe a edig almos cclic reeiio o olaio growh ad declie o eiher secies We make a mahemaical ormlaio o he above roblem Le be he rabbi olaio ad he corresodig olaio o oes a a give isa Sice here is a limied sl o carros he rabbi olaio grows as i he case o a irs order reacio relaive o he crre olaio Ths i he absece o oes d a a > I is aral o assme ha he mber o ecoers bewee oes ad rabbis is joil roorioal o heir olaios As hese ecoers will erich he o olaio b resls i he declie o rabbi olaio we d ma correc he above eqaio as a b where a b > I a similar d maer we obai c d where c d > Ths we have he ollowig o liear d ssem describig he olaios a b Volerra s re redaor eqaios Elimiaig we ge d c d The above eqaios are called a b d c d d The solio is a e b = K c e d where K = c a e d d or some iiial solio Drawig he grah is reall ogh ad Volerra has irodced a eicie aroach i his regard as discssed below We oe ha beig olaios are o egaive The lae is divided io 4 qadras ad he borderig ras are sed o rerese he osiive w direcios We ma ake = a e b ad w = K c e d Givig siable vales or ad ideedel lo he ad w grahs i he resecive qadras ad he obai he grah rom he w grah which is i ac he sraigh lie = w Diereial Eqaios 4

23 Noe ha d d gives = c/d ad = a/b called he eqilibrim olaios dx Le = X + c/d ad = Y + a/b The he ssem becomes dy dt Cosider he liearised ssem dx dy dt bc Y d ad X b bc Y bxy d ad X dxy b The solio o he liear ssem is a d X + b c Y = L a amil o ellises coceric wih he origi The grah rs o o be a oval abo he eqilibrim oi c/d a/b Diereial Eqaios 5

24 4 Aoomos ssems CHAPTER 4 NON LINEAR EQUATIONS Cosider he ssem d F d G -- Sice F ad G are ideede o he ssem is called aoomos The solio o he ssem is a air o cios describig a amil o crves i he - lae called he hase lae I is a mber ad is a give oi i he hase lae here eiss a iqe crve assig hrogh ad saisig he ssem Sch a crve is called a ah i he hase lae ad he lae wih all hese ahs will be called hase orrai o he ssem For a give ah we ma se orward arrows o idicae he direcio i which he ah is advacig as d o he ssem Sice A oi a which boh F ad G vaish is called a criical oi d ad a a criical oi o ah is assig hrogh a criical oi ad wo diere ahs will o iersec sice here eiss a iqe ah hrogh a give oi Give a aoomos ssem aar rom is solio we are ieresed i he locaio o he varios criical ois arrageme o ahs ear criical ois sabili o he criical ois ad he hase orrai Sabili : Le be a isolaed criical oi ad C = { } be a lim ah o We sa C aroaches as i ad C eers as i lim eiss or or I a ah C eers a criical oi he i aroaches i i a deiie direcio Eg Cosider he ssem d d -- The origi is he ol criical oi ad he geeral solio is obaied as = c e = c e + c e -- Whe c = we ge = = c e I his case he ah becomes he osiive or egaive ais accordig as c > or < ad as oi he ah aroaches ad eers he criical Diereial Eqaios 6

25 Whe c = we ge = c e ad = c e For c < he ah is he ra = < ad or c > he ah is he ra = > ad boh he ahs eers he criical oi as Whe c c eers as he he ahs are ½ - arabolas = + c/c Each o hese ahs Eg Cosider he ssem d d The ol criical oi is We obai he geeral solio as = c e + c e = c e - + c e Whe c = we ge = c e = c e I his case he ah becomes he ½ lie = ad as he ah aroaches ad eers he criical oi Whe c = we ge = c e - ad = c e - The ah is he ½ - lie = ½ < ad boh he ahs eers he criical oi as Whe c c he he ahs are disic braches o herbolas = C wih asmoes beig = ad = ½ ad oe o hese ahs aroaches he criical oi as or as Eg3 Cosider he ssem d d The ol criical oi is We obai he geeral solio as = - c si + c cos = c cos + c si which are circles wih commo cere All ahs are closed oes ad each o hem ecloses he criical oi ad oe o hese ahs aroaches he criical oi Eg4 Cosider he ssem d d The ol criical oi is B chagig o olar coordiaes we ge amil o sirals r = c e clock wise ashio o iii d We have Sabili ad asmoic sabili dr d r which gives he geeral solio as he so ha as he siral wids i he ai Cosider he aoomos ssem d F d G -- For coveiece assme ha is a isolaed criical oi o he ssem This criical oi is said o be sable i or each R > give here eiss r R sch ha ever ah which is iside he circle ceered a wih radis r or some = remais iside he circle ceered Diereial Eqaios 7

26 wih radis R or all > which is eqivale o sa ha ahs which ges siciel close o he criical oi sa close o i i is de corse ie as The criical oi is said o be asmoicall sable i here eiss a circle wih cere ad radis r sch ha a ah which is iside his circle or some = aroaches he cere as 4 Tes o Criical ois ad sabili o liear ssems Cosider he homogeeos liear ssem wih cosa coeicies d a b d a b assmig ha a b a b --- which has evidel he origi as he ol criical oi b Le = A e m = B e m --- be a o rivial solio whe ever m a + b m + a b a b = ---3 called he ailiar eqaio o he ssem Le m ad m be he roos o 3 We ma disigish he ollowig 5 cases Major cases: The roos m ad m are real disic ad o he same sigs The roos m ad m are real disic ad o oosie sigs 3 The roos m ad m are comle cojgaes b o re imagiar Border lie cases : 4 The roos m ad m are real ad eqal 5 The roos m ad m are re imagiar Case : The criical oi is called a ode The geeral solio is c Ae cbe m m c A e c B e a The roos m ad m are boh egaive m m Frher assme or recisio ha m < m < Whe c = we ge = c A e m = c B e / = B/A which eers he criical oi as he same lie which also eers as ---4 m --5 I c > he we ge ½ o he lie ad or c < we ge he oher ½ o Diereial Eqaios 8

27 Whe c = we ge = ca e m ad = c B e m 6 For c < he ah is ½ o he lie / = B/A which eers he criical oi as o he same lie which also eers as Whe c c also aroaches as ad or c > we ge he oher ½ he he ahs are crves Sice m ad m are boh egaive hese ahs m - m < each o hese ahs eers as Cosiderig he eressio or / rom 4 ad sice Noe ha / B/A as The criical oi is reerred as a NODE ad i his case i is asmoicall sable I m < m < he he above coclsio holds good wih a chage each crviliear ah eers alog he direcio B/A b The roos m ad m are boh osiive The siaio is eacl he same b all he ahs are aroachig ad eers as Assme m < < m The wo ½ lie ahs rereseed b 5 eer as rereseed b 6 eer as he wo ½ lie ahs B oe o he crviliear ahs rereseed b 4 corresodig o c c aroaches as or ; each o hem is asmoic o oe o he ½ lie ahs The criical oi is called a SADDLE POINT which is alwas sable 3 Le m a ib m a ib where a The geeral solio is Sose a < As e e a a [ c A [ c B cos b A cos b B si b c si b c B A si b A si b B cos b] --8 cos b] all ahs aroach b he do o eer i ad he wid arod i i a siral like maer Chagig o olar coordiaes d d/ d / a b a b -- 9 d > or < i a > or < Noe ha he discrimia D o he ailiar eqaio is egaive i he rese coe For = 9 gives d d =a Ths whe a > he > which imlies ha as all ahs siral abo i he ai clock wise sese The sese will be clock wise i a < The criical oi is called a siral which is asmoicall sable Diereial Eqaios 9

28 I a > he siaio is he same ece ha all ahs aroach as i is a sable siral 4 Le m = m = m sa Assme m < ad hece a a= b a = b = Le he commo vale be a The he ssem redces o d a d a ad is geeral solio is varios sloes ad sice m < each ah eers as The criical oi is a asmoicall sable border lie ode I m > he all ahs eer as ode b All oher cases Assme m < c Ae The geeral solio is cbe m m c c = c e m = c e m The ahs are ½ lies o The criical oi is a sable border lie A B A e B e Whe c = we ge he wo ½ lie ahs lig o / = B/A Sice m < boh o hem eer as I c he ahs are crviliear ad all o hem eer he criical oi as kee ageial o / = B/A as he aroach The criical oi is agai a asmoicall sable border lie ode I m > he i is sable 5 We ma reer case 3 wih a = Sice he eoeial acor is missig rom he solio he redce o eriodic cios ad each ah is closed srrodig he origi The ahs are acall ellises The criical oi is called a CENTRE which is sable b ca o be asmoicall sable We ma smmarise some o he observaios we have made i he seqece o he above discssio abo sabili Theorem The criical oi o he liear ssem is sable i boh he roos o he ailiar eqaio have o osiive real ars ad i is asmoicall sable ii boh roos have egaive real ars Takig = - m + m ad q = m m we ca reormlae he heorem as Theorem The criical oi o he liear ssem is asmoicall sable i ad q are boh osiive m m Diereial Eqaios 3

29 43 Liaov s direc mehod I a hsical ssem i he oal eerg has a local miimm a cerai eqilibrim oi he i is sable This coce leads o a owerl mehod or sdig sabili roblems Cosider he aoomos ssem d d F G Assme ha is a isolaed criical oi o he ssem Le C = [ ] be a ah Le E be a cio ha is coios ad havig coios irs arial derivaives i a regio coaiig C I is oi o C he E is a cio o aloe sa E Is rae o chage as he oi moves alog C is de E d E d E E = F G Le E be a coios cio wih coios irs arial derivaives i some regio coaiig he origi I E = ad he i is said o be osiive deiie i E > or ad egaive deiie i E < or semi-deiie i E = ad E or = ad E or Fcios o he orm Similarl E is called osiive ad egaive semi-deiie i E a m + b where m & are osiive iegers ad a & b are osiive cosas are osiive deiie Noe ha E is egaive deiie i E is osiive deiie; cios m or are o osiive deiie Give he liear ssem a osiive deiie cio E sch ha he derived cio E E H = F G is egaive semi-deiie is called a Liaov cio or B he earlier discssio we ge ha alog a ah C ear he origi de/ decreasig alog C as i advaces ad hece E is Theorem I here eiss a Liaov cio E or he ssem he he criical oi is sable Frhermore i his cio has he addiioal roer ha he derived cio H is egaive deiie he is asmoicall sable Proo: Le C be a circle o radis R > ceered a he origi ad i ma be assmed ha C is small eogh ha i is coaied i he domai o deiiio o E Sice E is coios ad osiive deiie i has a osiive miimm m o C Sice E is coios a he origi ad vaishes here we ca id < r < R sch ha E < m wheever is iside he circle C o radis r ad ceered a he origi Le C be a ah which is iside C or = The E < m ad de/ imlies ha E E < m or Diereial Eqaios 3

30 all > I ollows ha he ah C ca ever reach he circle C or > Ths is sable Uder he addiioal assmio we claim rher ha E as iml ha he ah C aroaches as decreasig cio Sice E is boded below b E L sices o show ha L = This wold Now alog C de/ < E is a sa as The i Sose o Choose < r < r sch ha E < L/ wheever is iside he circle C3 wih radis r Sice H is egaive deiie i has a egaive maimm -k i he closed als boded b C ad C3 Sice his regio coais C or E = E + de which gives E E - k B sice righ side o he ieqali becomes egaivel iiie as Ths L = ad he roo is comlee E as This coradics he ac ha E Eg Cosider he eqaio o moio o a mass m aached o a srig d d m c k Here c is he viscosi o he medim hrogh which he mass moves ad k> is he srig cosa The eqivale aoomos ssem is d d k m The ol criical oi is The kieic eerg o he mass is m / ad he oeial c m eerg de o he crre elogaio o he srig is kd k Ths he oal eerg o he mechaical ssem is E = ½ m + ½ k The E is osiive deiie ad H = k + m -k/m -c/m = - c Ths E is a Liaov cio ad he criical oi is sable b Theorem E Show ha is a asmoicall sable criical oi o he ssem d 3 3 d 5 3 Le E = a m + b where a b > ad m are osiive iegers E is osiive deiie ad H = ma m b = - 6 ma m+ ma m- Diereial Eqaios 3

31 + b b + Le m = 3 = a = b = 3 The H = is egaive deiie Now E is a Liaov cio or he ssem wih he derived cio H egaive deiie Ths he criical oi is asmoicall sable E Show ha is a asmoicall sable criical oi o he ssem d d 3 3 Take E = + 44 Simle criical ois No liear ssem d F Cosider he aoomos ssem wih a isolaed criical oi a d G Y Sice F = = G assmig heir Maclari s series easios abo ad eglecig higher owers o & or close o he ssem redces o a liear oe d a b More geerall we ma ake he ssem as d a b g I is assmed ha a b a b so ha he criical oi will be isolaed lim is called a simle criical oi o he ssem i lim g ad Theorem Le be a simle criical oi o he o liear ssem d a b -- d a b g I he criical oi o he associaed liear ssem d a b -- d a b Diereial Eqaios 33

32 alls der a o he hree major cases Node Saddle oi siral he he criical oi o is o he same e Remark: There will o be similariies amog he ahs i boh he ssems I he o liear case ahs will have more disorios I ssem has he origi as a border lie ode cere he origi will be eiher a ode or siral cere or siral or he ssem Theorem Le be a simle criical oi o he o liear ssem ad cosider he relaed liear ssem I is a asmoicall sable criical oi o he i is asmoicall or Proo We ma cosrc a siable Liaov cio or he ssem o jsi he claim The coeicies o he ailiar eqaio o he liear ssem amel & q will be osiive b he assmio ha is asmoicall sable or Now deie E = ½ a + b + c a b ab ab where a = q a b ab ab c = ad b = - q a a bb q Noe ha = - a + b & q = a b a b We have q a b > ad i ca be direcl show ha ac b > Sice b ac < & a > E is osiive deiie I ca also be easil obaied ha H = a + b a + b + b + c a + b = - + which is egaive deiie Ths E is a Liaov cio or he liear ssem Now b sig he coii o & g a ad shiig olar coordiaes i ca be E E show ha F G where F = a + b + ad G = a + b + g is egaive deiie Ths E is a osiive deiie cio wih he derived cio relaed o he o liear ssem egaive deiie Hece b Theorem is asmoicall sable or he ssem Eg The eqaio o moio or damed vibraios o a simle edlm is d c / m gravi d g / asi --- where c > ad g is he acceleraio de o Diereial Eqaios 34

33 The eqivale aoomos ssem is d d g / asi c / m -- ca be wrie as This = ad g = g/a -si lim Sice d d g / a c / m g / a si g lim si --3 is a simle criical oi o he o liear ssem As or si si si / Now is a isolaed criical oi o he associaed liear ssem d d g / a c / m We have = c/m > q = g / a > or Hece b Theorem is asmoicall sable Ths b he las Theorem is a asmoicall sable criical oi o he origial ssem Sice = ad = d/ = reers o he mea osiio ad iiial veloci asmoic sabili o imlies ha he moio de o a sligh disrbace o he simle edlm will die o wih he assage o ime We give a ew more Theorems hell i he ivesigaio o a aoomos ssem Theorem A closed ah o a aoomos ssem ecessaril srrods a leas oe criical oi Ths a ssem wiho criical ois i a give regio ca o have closed ahs i ha regio F G Theorem I is alwas osiive or alwas egaive i a cerai regio o he hase lae he he ssem ca o have closed ahs i ha regio Diereial Eqaios 35

34 Proo: Assme ha he regio coais a closed ah C wih ierior R The b Gree s Theorem Fd Gd ha C R F G d d B alog C d = F & d = G so C Fd Gd = a coradicio Theorem3 Poicare-Bediso Le R be boded regio o he hase lae ogeher wih is bodar ad sose R does o coai a criical ois o he ssem I C is a ah ha eers R ad remais i R i is rher corse he C is eiher a closed ah or i sirals oward a closed ah as Ths i eiher case he ssem has a closed ah Theorem4 Lieard Le he cios ad g sais he ollowig codiios boh are coios ad have coios derivaives or all g is a odd cio sch ha g > or > ad is a eve cio 3 he odd cio F d has eacl oe osiive ero a = a is egaive or < < a is osiive ad o decreasig or d d > a ad F as The he Lieard s eqaio g has a iqe closed ah srrodig he origi i he hase lae ad his ah is aroached sirall b ever oher ah as Diereial Eqaios 36

35 5 Mehod o sccessive aroimaios CHAPTER 5 SOME FUNDAMENTAL THEOREMS Cosider he iiial vale roblem IVP = = -- where is a cio coios i some eighborhood o A solio is geomericall a crve i he lae ha asses hrogh so ha a each oi o he crve he sloe is rescribed as The IVP is eqivale o he iegral eqaio IE = + [ ] -- { is eqivale o : Sose is a solio o The is ideed coios ad i we iegrae i rom o is obaied I is a coios solio o he = ad b diereiaig = } We ma sgges a ieraive rocedre o solve he IE Sar wih he rogh aroimaio = Sbsiig i he righ side o we ge a ew aroimaio as = + [ ] Ne se i RS o o ge = + ] [ This rocess ca be coied o ge = + [ ] The rocedre is called Picard s mehod o sccessive aroimaios Eg Cosider he IVP = = Eqivale IE is = + The = + Wih = = + = + = + = + + / 3 = + / = 3 3 I is clear ha 3 3 Noe ha = e is a solio o he IVP e Diereial Eqaios 37

36 Eg Cosider = + = We ma ake = The = = 4 / 4 / = 4 / = / 4 = / + 5 / / = / + 5 / + 8 /6+ /44 Eg3 Cosider = + = I is o diicl o ge he eac solio as = e Wih = = + =! = + /! = 3 3 = 3! ! Noe ha + + e = e 5 Picard s Theorem Theorem Le ad be coios cios o a closed recaglar regio R wih sides arallel o he coordiae aes I is a ierior oi o R he here eiss a mber h > wih he roer ha he iiial vale roblem = = --- has a iqe solio = i [ h + h ] Proo: We kow ha ever solio o he IVP is also a coios solio o he IE = + [ ] --- ad coversel We will show ha has a iqe solio i [ h + h ] or some h > We ma a rodce a seqece o cios ollowig Picard s mehod o sccessive aroimaios Le = = + [ ] = + [ ] = + [ ] Diereial Eqaios 38

37 Claim The seqece < > coverges o a solio o he IVP i [ h + h ] or some h > Sice R is comac ad ad are coios i R he are boded Thereore here eiss M K > sch ha M --3 ad K 4 R Le be disic ois i R * The b Mea vale heorem --5 or some * bewee & The b 4 we ge K --6 Now choose h > sch ha K h < --7 ad he recaglar regio R deied b h ad Mh is coaied i R Sice is a ierior o R sch a h eiss Noe ha is he h arial sm o he series --8 Ths o show ha < > coverges i is sicie o show ha he series * coverges a The grah o he cios = or ever This is clear or = = h lies i R ad hece i R or Sice are i R we ge [ ] Mh Ths grah o = lies i R Now i rs o ha [] are i R ad [ ] Mh Ths grah o = lies i R Proceedig similarl we ge he resla Sice is coios i h here eiss a cosa a = ma Sice [] [] are i R 6 K Ka ad [ ] Kah =akh Similarl K K Kah K ah so Diereial Eqaios 39

38 3 [ ] K ah h a Kh Coiig like his we ca show ha a Kh Now each erm o he series is domiaed b he corresodig erm o he series + a + a Kh + a Kh + which coverges beig esseial a geomeric series wih commo raio r = Kh mericall less ha b 7 Ths b comariso es he series 8 coverges iorml i h o a sm sa ad hece < > coverges o iorml i h Sice he grah o lies i R he grah o he limi cio also lies i R Sice each is coios he iorm limi is also coios Now we roceed o rove ha is a solio o he IVP ie We have o show ha - - [ ] = --9 B - - [ ] = Now cosider - - [ ] - = - - [ ] - [ - - [ ] ] = [ ] + [ ] [ ] + [ ] [ ] + K h ma [ - ] sice grah o lies i R ad hece i R ad b vire o 6 The iorm covergece o o will eable s o make he righ side o he above ieqali as small as we lease b akig siciel large Sice he le side is ideede o we ge he reqired resl Now we sele he iqeess ar Le be aoher ossible solio o he IVP i h I is esseial o show ha he grah o also lies i R O he corar assme is grah leaves R The here eiss some i h sch ha Mh ad he Diereial Eqaios 4

39 coii o a = will give Mh i The Mh Mh / h M where as b mea vale heorem here eiss * bewee & sch ha * * * M sice * * lies i R Hece a coradicio Sice boh & are solios o he IVP [ ] Kh ma sice grah o boh cios lie i R So ma sice K h < K h ma B his will iml ma = Ths we have = or ever i he ierval h Remark We oice ha he coii o is made o se o i he roo o he ee ha i imlies 6 We ca relace his reqireme b a Lischi s codiio amel here eiss K > sch ha K I we rher dro his codiio oo i is kow ha he IVP sill has a solio b he iqeess ca o be asceraied Peao s Theorem ' Cosider he IVP 3 Le R be he recaglar regio Here = 3 / is coios i R Two diere solios are = 3 ad = Theorem Le be coios ad sais he Lischi s codiio K o a verical sri a b ad I is a oi o he sri he he iiial vale roblem = = has a iqe solio i [ a b ] Proo: The roo is similar o ha o Theorem ad based o Picard s mehod o sccessive aroimaios Le M = M = ma M = M + M I ca be easil observed ha M ad M Diereial Eqaios 4

40 Diereial Eqaios 4 Assme b The ] [ K KM 3 ] [ K / M K M K ad i geeral! / M K The same resl holds or a b has o be relaced b Ths we have! / M K! / a b M K Now each erm o he series * is domiaed b he corresodig erm o he coverge merical series! a b M K a b KM M M ad hece he series coverges iorml i [ a b ] o a limi cio The iorm covergece will readil iml ha is a solio o he IVP i [ a b ] I ossible le be aoher solio o he IVP We claim ha he also so ha = Now is coios ad = + ] [ Le A = ma The or b ] [ KA K ] [

41 K K A K A! ad i geeral K A! A similar resl is go or a Ths or a i [ a b ] K A! K b a A! B rom eoeial series we ge or a r r /! as Ths he righ side o he above ieqali eds o ero as Hece we ge Remark also Picard s mehod o sccessive aroimaios ca be alied o ssems o irs order eqaios b sarig wih ecessar mber o iiial codiios b coverig io a ssem o iegral eqaios Picard s heorem der siable hohesis holds good i his coe also Theorem Le P Q ad R be coios cios i[ a b ] I is a oi i [ a b ] ad are a wo mbers he he iiial vale roblem d P d i [ a b ] d d Q R = ad = has a iqe solio = Diereial Eqaios 43

42 CHAPTER 6 FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 6 Irodcio - Review Parial diereial eqaios arise rom a coe ivolvig more ha oe ideede variable For he aalsis o a arial diereial eqaio ad is solio geomericall we reqire good kowledge abo rereseaio o crves ad sraces i 3 dimesio A crve C i 3 dimesio ca be seciied i arameric orm as he collecio o ois saisig he eqaios = = = 3 where he arameer varies i a ierval I i R ad 3 are coios cios o I The sadard arameer is he arc legh s rom a ied oi A o C o a geeric oi P o C Eqaio o C ca also be reseed i vecor arameric orm as r = i + j + 3 k A a oi P r o C dr gives a age direcio I s is he arc legh arameer he dr ds gives he i age direcio Eg A sraigh lie wih direcio raios l m ad assig hrogh a b c ca be wrie i smmeric orm as l = m = Eg A righ circlar heli o he clider + = a ca be seciied as = acos = a si = k Eqaio o a srace is sall ake as F = where F is a coiosl diereiable cio i R 3 Is eqaio ca also be eressed i arameric orm as F F = F v = F v = F3 v I he ad v ca be solved as v cios o ad locall sa = v = The rom he las eqaio we ge = F3 Sose he crve C : = s = s = s lies o he srace S whose eqaio is F = The Fs s s = or ever s Diereial Eqaios 44

43 O diereiaig w r s we ge F d ds + F d ds + F d ds = This imlies ha a he oi P o he crve he direcio F F F is erediclar o he age direcio Sice C lies eirel o S he above is a age direcio o he srace also Ths F F F is a ormal direcio o S a P d ds d ds d ds o C Le be a variable deedig o he ideede variable The a eqaio o he orm = where is a cio wih coios arial derivaives o order o or some is called a arial diereial eqaio PDE o order The order o a PDE is he order o he highes order derivaive aearig i i A PDE is said o be qasi liear i i is liear i is highes order derivaives; ad semi liear i i is qasi-liear ad he coeicies o he highes order derivaives does o coai he deede variable or is derivaives A PDE which is o qasi liear is called o-liear A semi-liear PDE which is liear i he deede variable ad is derivaives is called liear A he begiig we ma cosider he case o ol wo ideede variables sa ad ad oe deede variable sa deedig o ad We ma se he oaios = q = r = s = = I his coe he irs order PDE has he orm q = We reqire he ollowig Theorem i Real Aalsis i ma coes which ivolves solvig oe se o variables i erms o he ohers rom a give cioal eqaio Imlici Fcio Theorem Le be a coiosl diereiable cio rom a oe se E o R +m o R Le a b E sch ha a b = Le A = a b I A is iverible he here eiss W a eighborhood o b i R m sch ha or each i W here eiss a iqe i R sch ha = The Theorem gives he logical sor i idig i erms o give = Diereial Eqaios 45

44 Diereial Eqaios 46 6Formaio o Firs Order PDE Cosider a amil o sraces o he orm F v = where F is a arbirar cio ad ad v are give cios o Diereiaig F v = ariall w r we ge F + v F v v = F q + v F q v v = O elimiaig F ad v F rom & we ge q v v q v v = 3 which ca be simliied as v v + v v q = v v 3 or i erms o Wroskias v + v q = v 3 which is a qasi liear eqaio Ths v + v q = v is he PDE associaed wih he amil o sraces F v = where F is a arbirar cio ad & v are give cios o Remark: Le a v be cios o ad I v ca be eressed as a cio o aloe wiho ivolvig ad he v = Here we sa ad v are cioall deede Le v = H where H is some cio The v = H & v = H Elimiaig H we ge v = v v = Now cosider a wo arameer amil o sraces = F a b where a ad b are arameers Diereiaig ariall w r ad we ge = F a b ad q = F a b 3

45 Sose he mari F F a b F F a b Fa Fb is o rak The b Imlici cio heorem we ca solve or a ad b rom wo o he above hree eqaios i erms q ad sbsiig i he remaiig eqaio we ge a PDE q = Eg Cosider = + a + b Diereiaig wr & resecivel = + a ad q = a 3 Elimiaig a bewee & 3 we ge q = a PDE Eg Elimiae a ad b o orm a PDE give = a + b Diereiaig ariall wr ad we ge = a & q = b The PDE is obaied b elimiaig a ad b rom he above eqaios Ths = + q Eg3 Elimiae he arbirar cio F o orm he PDE o he amil o sraces = + + F O diereiaio = + F & q = + F 3 Elimiaig F rom eqaios & 3 we ge he PDE q = Eg4 Elimiaig F orm he PDE o he oe arameer amil o sraces F + - = Le = + v = - O diereiaio wr F + + Fv / = ad F + q + Fv -/ eqaios q = Elimiaig F ad Fv bewee he las wo We ge he PDE / / q = ie -/ q - + / = ie q = E Form he diereial eqaio give a = + F + b F = c = F / E Form he diereial eqaio give = + a + b = a + + b 3 + a 3 = + a + b 3 We have he classiicaio o irs order PDEs as give below Liear Eqaio P + Q q = R + S Diereial Eqaios 47

46 Semi-liear eqaio P + Q q = R 3 Qasi-liear eqaio P + Q = R 4 No-liear eqaio q = The solio o a irs order PDE i is a srace i 3 dimesio called a iegral srace There are diere classes o iegrals or a give PDE 63Classiicaio o Iegrals Cosider he PDE q = a Comlee iegral A wo-arameer amil o solios o he PDE q = * = F a b is called a comlee iegral o * i i he regio o deiiio o he PDE he rak o he mari F F a b F F a b F F a b is This codiio imlies ha a leas oe o he sb marices F F a b F F a b F F a b F F a b Fa Fa F F is o-siglar ie iverible I garaees ha we ca solve or a ad b rom wo o he eqaios = F a b = F a b ad q = F a b 3 i erms or or q ad he elimiaio o a ad b b sbsiig i he remaiig eqaio so ha eqaio * is recovered or saisied This is a coseqece o Imlici Fcio Theorem b Geeral iegral Le = F a b be he comlee iegral o * where a ad b are arbirar cosas reerred as arameers i he geomerical coe ha rereses a woarameer amil o sraces i 3 dimesio b b Diereial Eqaios 48

47 Le s assme ha a ad b are cioall relaed so ha b = a where is a arbirar cio Corresodigl we ge a oe-arameer sbamil = F a a o he woarameer amil o sraces rereseed b The eveloe o his amil i i eiss is also a solio o he PDE * called he Geeral Iegral The eveloe is obaied b elimiaig he arameer bewee he eqaios ' = F a a ad = F F a 3 obaied b diereiaig ariall w r he arameer a The elimiaio will give G = a srace i 3-dimesio a I isead o a arbirar cio we se a deiie relaio bewee a ad b like b = a or b = a + a or b = si a ec ad roceedig o id he eveloe o he corresodig sb-amil o he he reslig eveloe i i eiss is a solio o * called a ariclar iegral c Siglar Iegral The eveloe o he wo-arameer amil o sraces = F a b i i eiss is also a solio o he PDE * called he siglar iegral The eveloe ca be obaied b elimiaig he arameers a ad b rom he eqaios = F a b = Fa a b ad = Fb a b 3 d Secial Iegral I cerai cases we ca obai solios which are o allig der he above classes called Secial Iegrals For he PDE q = solios meioed above b = is solio which is o belogig o he hree class o Theorem The eveloe o a -arameer amil = F a o solios o he PDE q = i i eiss is also a solio o he PDE Proo: The eveloe is obaied b elimiaig he arameer a bewee = F a ad = Fa a Ths he eveloe is = G = F a where a is obaied rom b solvig or a i erms o ad For ois o he eveloe G = F + Fa a = F ad G =F + Fa a = F sice o he eveloe Fa = Ths he eveloe has he same arial derivaives as hose o a member Diereial Eqaios 49

48 o he amil a a give oi Sice he PDE a a oi is a relaio o be saisied b he coordiaes o he oi ad he arial derivaives a ha oi we ge ha he eveloe is also a solio o he PDE Theorem The Siglar iegral is a solio Proo: Le = F a b be he comlee iegral o q = * The siglar iegral o * is obaied b elimiaig a ad b bewee = F a b = Fa a b = Fb a b 3 Hece he eveloe is = G = F a b where a & b are obaied rom & 3 b solvig or a & b i erms o & For he eveloe G = F + Fa a + Fb b = F ad G = F + Fa a + Fb b = F sice Fa = Fb = o he eveloe Ths a a oi o he eveloe he arial derivaives will be he same as a member o he amil Hece he eveloe is also a solio o * Remark : Recall ha a eveloe o a amil a give oi o i oches a member o he amil Remark: The siglar iegral ca also be deermied direcl rom he give PDE * b he ollowig rocedre The siglar solio is obaied b elimiaig ad q rom he eqaios q = * q = ** q q = *** reaig & q as arameers Le = F a b be he comlee iegral o * The F a b F a b F a b = which holds or ever a & b I ca be diereiaed ariall wr a & b so ha Fa + Fa + q Fa = ad Fb + Fb + q Fb = Sice o he siglar iegral Fa = & Fb = he above eqaios will simli o Fa + q Fa = ad Fb + q Fb = # Sice he mari F F a b F F a b F F a b Fa Fa is ad Fa = & Fb = we ge o- F F b b siglar Hece # gives = & q = Hece he resl Eg I ca be show ha = F a b = a + b + a + b is a comlee iegral o he PDE q = - q q = From b diereiaig ariall wr & we ge = a & q = b Diereial Eqaios 5

49 The eqaio is saisied b ie is a solio o or a wo arbirar cosas a & b Frher F F a b F F a b F F a b a = b Ths is a comlee iegral o We ca id ariclar iegrals b relaig corresodig sb-amil Le b = a is o rak a & b ad idig he eveloe o he The we ge he amil = a + + a Diereiaig w r a = a O elimiaio o a we ge he eveloe as = - + / /4 = - + / 8 or = a ariclar iegral The siglar iegral is obaied as ollows Elimiae a & b rom = a + b + a + b = + a = + b The siglar iegral is = Remark: A PDE ca have more ha oe comlee iegral so ha he erm Comlee ma o be misierreed ad he ariclar iegrals or he siglar iegral are o members o he amil = F a b or some vales o a & b 64Liear eqaios The ollowig Theorem rovides a mehod or idig he Geeral iegral o a give qasi liear eqaio Theorem The geeral iegral o he qasi liear eqaio P + Q = R where P Q R are coiosl diereiable cios o ad is F v = where F is a arbirar cio ad ad v are cios sch ha = c ad v = c are wo ideede solios o he ssem o ordiar diereial eqaios d P d Q d R Proo: Sice = c is a solio o 3 d = d + d + d = ad hece P + Q + R = 4 Similarl we ge v P + v Q + v R = 5 Ths rom eqaios 4 & 5 P v = Q v ha ad v are ideede = R v 6 Here we se he assmio Diereial Eqaios 5