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1 Power Series Methods Natioal Chiao Tug Uiersit Chu-Je Tsai /6/

2 Power Series A power series i a is a series of the form a Suh a series is said to be etered at a. A power series is oerget at a alue I if the limit of partial sums eists, i.e. f lim N N eists. The iteral of oergee, I, is the set of all real umber where the series oerges.. a /68

3 3/68 Power Series Represetatio of f B Talor a or Malauri a series epasios, a futio f a be writte i power series forms: L L L L ! l 5! 3!! si 3!!! e

4 Power Series Operatios If the Talor series of a futio f a a! oerges to f for all i some ope iteral otaiig a, the we sa that the futio f is aalti at a. Polomials are aalti; a ratioal futio is aalti whereer the deomiator is ot zero. If f ad g are aalti at a, so are fg, f g, ad f/g if ga. The operatios of additio, multipliatio, ad diisio a be applied o power series as i polomials. 4/68

5 5/68 Eample: Addig Two Power Series Write as oe power series. Solutio: 3. ] [

6 Power Series Method The power series method for solig a DE osists of substitutig the power series i the DE ad determiig the oeffiiets,, so that the equatio satisfies. If f, the f. If a b, for all i the iteral of oergee, the a b for all. 6/68

7 7/68 Eample: ' Sie we hae Perform hage of ide to alig, We hae a reurree relatio /,. /!,,, ad. [ ]..!

8 Radius of Coergee A power series f. has a radius of a oergee ρ. If ρ >, f oerges for a < ρ, ad dierges for a > ρ. If f oerges ol at its eter a, the ρ. If it oerges for all, ρ. Gie the power series g, if the limit eists, the ρ lim If ρ, the g dierges for all. If < ρ <, g oerges if < ρ, ad dierges if > ρ. If ρ, the g oerges for all. 8/68

9 9/68 Eample: ' / Sie the Therefore,, 3,!,.,, ad....!

10 Eample: ' / Now, let s he the radius of oergee of this series:! ρ lim lim.! The series oerges ol for. This meas that there is o power series solutio to the origial DE! /68

11 Power Series Solutios Suppose the liear d -order DE A " B' C is put ito the stadard form the: " P' Q, A poit a is said to be a ordiar poit of the DE if both P ad Q are aalti at a. A poit that is ot a ordiar poit is said to be a sigular poit of the equatio. /68

12 Eample: Ordiar, Sigular Poits Eer fiite alue of is a ordiar poit of " e ' si. is a sigular poit of the DE " e ' l. /68

13 Polomial-Coeffiiet DEs Reall that a polomial is aalti at a alue, ad a ratioal futio is aalti eept at poits where its deomiator is zero. Eamples The Euler equatio a b' has a sigular poit at. The equatio has sigular poits at ±i. 3/68

14 Solutios Near a Ordiar Poit Theorem: If a is a ordiar poit of A B C, we a alwas fid two liearl idepedet solutios i the form of a power series etered at a, that is, a A series solutio oerges at least o some iteral defied b a < ρ, where ρ is the distae from a to the earest sigular poit.. 4/68

15 5/68 Eamples: 4 3, Note that the ol sigular poits are ±, there should be a series solutio with radius of oergee. Sie Therefore, Colletig oeffiiets of ad, we obtai,, ad. 3 4.,

16 6/68 Eamples: 4 3, / Whe,, 4,, we hae Whe, 3, 5,, we hae Therefore, the solutio is ad 4, , , 4 4 3, L L , , , L L. 3!! L L

17 Traslated Series Solutios If the iitial oditio is gie at a other tha zero, we hae to assume the geeral solutio form a. This wa, we a obtai the IVP solutio with a ad a. 7/68

18 Eample: d t t 3 dt 3 t d dt, 4 We a tr the solutio form Σ t. Or, we a perform a hage of ariable to the DE b t : ad d dt t t 3 3 4, d d d ad d dt d dt Hee, the DE beomes 4 3 Same DE as the preious oe. Substitutig t ito preious solutio, we get 3 3 t 4 t t t t 6 3 whih oerges if < t < 3. d, d d d d 4 d dt L,. 8/68

19 9/68 Eample: / Agai, assume Σ, we hae Therefore,, 3 /6, ad ad determie the whole sequee of, 4.,. [ ]. 6 3., alig

20 Eample: / We a fid two liearl idepedet solutios ad b first hoosig ad, the hoose ad. Thus, ad L Note that ad are liearl idepedet. The geeral solutio is a liear ombiatio of ad. 7, L. /68

21 /68 Eample: os / Sie 6 4 6! 4!! os L L L L !! L

22 Eample: os / We hae:, 63, 4, L Therefore: L, L 6 3 with regio of oergee < a Plot of s b Plot of s. /68

23 Legedre s Equatio Legedre s equatio of order α is the d -order liear DE of the form αα, Where the real umber α satisfies the iequalit α >. The ol sigular poits of the Legedre s equatio are at ad. 3/68

24 Solutio of Legedre s Equatio / Sie is a ordiar poit of the equatio, substitute Σ m m ito Legedre s equatio, we hae α α It a be show that, α α α α ad m m m, m m m m m 4 L m α α 3 L α m, m! m m α α 3 L α m α α 4 L α m. m! m. 4/68

25 Solutio of Legedre s Equatio / If α, a o-egatie iteger, we hae ad 3! 4! ! L ! 5! ! Notie that if is a ee iteger, termiates. Whe is a odd iteger, termiates L 5 5/68

26 6/68 Legedre Polomials / The solutio polomial of Legedre equatio of order, with speial seletio of ee or odd, are alled Legedre polomial of degree : where N /. For eample: , , 3, P P P P P P 4,!!!! N P

27 Legedre Polomials / The are the solutios of : : : 3: M M 6.5 P P Legedre Polomials, for,,, 3, 4 7/68

28 Solutios aroud Sigular Poits Let Σ, if P is ot aalti at, P Σb will ot oerge at. Howeer, it is possible that ma still oerge. P Σb Σ I geeral, ee if is a sigular poit, the power series epressio of P Q ma still oerge to zero. 8/68

29 Regular Sigular Poits Assume that a DE i the stadard form P Q has a sigular poit at. If there eists two futios p ad q, both are aalti at, suh that p P ad q Q, the origial DE a be put ito the form: p q, the, we all a regular sigular poit of the DE. Otherwise, is a irregular sigular poit. 9/68

30 Remars o Sigularit of P ad Q If is a sigular poit, the power series epasio of P at approahes. Howeer, if P grows slower tha / whe, the P is oerget. That is, p P is aalti at. Similarl, q is aalti at if Q grows slower tha /. Note that, for the DE p q, if is a sigular poit, it is regular if p ad q are polomials. 3/68

31 Eample: 4 3 The stadard form of the DE is 4 3 The DE a also be epressed as. 4 / 3 /. Therefore, is a regular sigular poit sie p ad q are both aalti at. 3/68

32 Eample: No-zero Sigular Poits For 4 3 5, ad are sigular poits. We hae P 3 ad 5 Q Obiousl is a regular sigular poit, ad is a irregular sigular poit. The DE a be put ito the form b t :. 3 t 4 t 5 t 4 t. 3/68

33 Eample: No-polomial p, q The DE 4 si os a be epressed as si / os /. Sie is ot a ordiar poit ad p si 3 3! 4 L L, 3! 5! are both aalti oerget at, thus is a regular sigular poit. 5 5! 4 os q L! 4!! 4! 4 6! L, 33/68

34 Solutio ear Sigular Poits If we hae a ostat-oeffiiet Cauh-Euler equidimesio equatio: p q, where p ad q are ostats, the r is a reasoable guess of the solutio r is a root of the equatio: rr p r q. If we hae oeffiiet futios p ad q istead, is it possible that r r is a solutio? 34/68

35 35/68 Method of Frobeius If is a regular sigular poit of the DE a a a, the there eists at least oe solutio of the form where the umber r is a ostat to be determied. The series will oerge at least o some iteral < < R., r r

36 Eample: 3 /3 Solutio: let r, we hae Therefore, 3 r r3r We hae: r r ad r r r [ r 3 3r ] r3r r 3 3r,,,, L. 36/68

37 Eample: 3 /3 Hee, r, / 3,,,, L r 3 3r Substitutig r ad r /3 ito the reurree eq., r / 3, r,!5 8 L3! 4 7L3 37/68

38 38/68 Eample: 3 3/3 Let, we hae two series solutios Sie ad are liearl idepedet o the etire ais, is the geeral solutio of the DE o a iteral ot otaiig the origi ote that is udefied ! 3 8!5 /3 L L

39 Idiial Equatio The equatio that ostrais the epoets r i the Frobeius theorem is alled the idiial equatio. The solutios of the idiial equatio are alled the idiial roots. Whe is a regular sigular poit, the idiial equatio is obtaied b substitutig Σ r ito the DE ad equatig the total oeffiiets of the lowest power of to. 39/68

40 Frobeius Series Solutios / Theorem: Suppose that is a regular sigular poit of p q. Let ρ > deote the miimum of the radii of oergee of the power series p p ad q Let r ad r be the real roots, with r r, of the idiial equatio rr p r q. The, q. 4/68

41 Frobeius Series Solutios / a For >, there eists a solutio of the form r a a orrespodig to the larger root r. b If r r is either zero or a positie iteger, the there eists a seod liearl idepedet solutio for > of the form: r b b orrespodig to the smaller root r. The radii of oergee of the solutios are at least ρ. 4/68

42 Eample: Sie ½ ½, we hae p / ad q p ½ ad q rr r/, r, ½. For r ½, substitute a r ito the DE, we hae a a! For r, substitute b r ito the DE, we hae b b L. 4/68

43 Eample: This DE is the Bessel s equatio of order zero to be disussed i detail later. Sie p ad q, we hae the idiial equatio r. The ol root is, ad the solutio falls ba to the power series form: The reurree relatio beomes /,. The solutio is the Bessel futio of the first id: J! The seod solutio will be disussed later... 43/68

44 44/68 Eample: /3 I geeral, whe r r is a positie iteger, the Frobeius series solutio ol eists for r the larger root. Howeer, i this eample, we hae two solutios. The DE a be writte as The idiial equatio rr r has roots,. Start with r, let Hee,...

45 Eample: /3 The first two terms gies us ad, whih meas ad a be arbitrar ostats. Thus, the reurree relatio /, a be diided ito two groups of oeffiiets: ad, for!! Therefore, a geeral solutio is!!.. 45/68

46 Eample: 3/3 Now, if ou pa attetio, ou will reogize that the solutio is simpl os si. The graph of the solutio is: 46/68

47 Eeptioal Case of r r N Whe r r N ad N Z, the Frobeius form based o r would be r N N r N, where is the solutio orrespodig to r. If is substituted ito p q, It ma ot be possible to fid a solutio for N. Note, the oeffiiet futio F r of r has the form: F r φr L r;,,,, where φr has the idiial equatio form, ad L is a liear ombiatio of,,, for eah. 47/68

48 Eamples For 6, r ad r 5, the Frobeius form of r leads to the reurree relatio 5 5, whih still gies a alid idetit at 5. So the problem has d solutio i Frobeius form. For 8, r 4 ad r, the Frobeius form of r leads to the reurree relatio 6, whih fails whe 6 beause aot be zero sie, if, we will obtai the triial solutio. 48/68

49 d Solutio b Redutio of Order If there is ol oe solutio i Frobeius form for P Q, we a fid the seod solutio b redutio of order. That is, let the seod solutio be u, ad substitute it ito the DE to fid a u that satisf the equatio. Reall that the redutio of order formula tells us P d ep d. 49/68

50 Summar of Idiial Roots / Case I: r ad r are distit, r r N, for some iteger N eists two liearl idepedet solutios of the form r. Case II: r r N, for some iteger N eists two liearl idepedet solutios of the form C r, l r b, b Note that C ould be zero.. 5/68

51 5/68 Summar of Idiial Roots / Case II otiued: If C, the d solutio a be obtaied b: Case III: If r r, there eists two liearl idepedet solutios of the form. d e d P. l, r r b

52 Bessel s Equatios Bessel s equatio of order is defied as. The solutios are alled Bessel futios of order. The futio first appears i 764 whe Euler was studig the ibratio of drum membrae. Later, the futios appears i ma phsis problems, from fluid equatios to plaet motios. 5/68

53 Reiew o Gamma Futio Γ The gamma futio is defied as Γ dt. For >, we hae Γ Γ. t e t 53/68

54 54/68 Solutio of Bessel s Equatio / Let the solutio be, we hae The idiial equatio is r, pi r r [ ]. r r r r r [ ].

55 55/68 Solutio of Bessel s Equatio / Therefore, we hae,...,,, Γ Γ Γ Γ if,!! 3,...,,,! L L

56 56/68 Bessel Futios of the st Kid / The solutios of Bessel s Equatio a be writte as Similarl, startig from r, we hae J ad J are alled Bessel s futios of the first id of order ad. Γ! J.! Γ J

57 Bessel Futios of the st Kid / Now, we wat to fid the geeral solutio of the Bessel s DE. Notie that r r :. If iteger, the J ad J are liearl idepedet.. If m, m is a iteger, the J m/ ad J m / are still liearl idepedet. 3. If m, m is a iteger, the J m ad J m are liear depedet solutios of Bessel s DE. must fid aother solutio! 57/68

58 J m & J m are Liearl Depedet / Proof: Assume that m is a iteger, we wat to show that J m m J m. Perform hage of ide o J m : J m! Γ m m Let m m m ad m, we hae J m m m m! Γ m 58/68

59 59/68 J m & J m are Liearl Depedet / Sie Γ, for,,,, we hae 3 Fiall, ote that m!γ [ mm ]! Γ! [mm ] Γ! Γ m. Therefore, Γ! m m m m J! J m J m m m m m Γ #

60 Eample: ¼ Sie the DE a be rewritte as ½, we hae ½. The geeral solutio is J ½ J ½ J J Bessel futios of the first id,,,, 3, 4 6/68

61 6/68 Bessel Futios with Whe, we hae J J, the d solutio a be obtaied b Case III of the method of Frobeius: J, ad Substitute ito the DE ad sole for b, we hae: γ is Euler s ostat.. l! b! l J L π γ π

62 Bessel Futios of the d Kid If is a real alue, we a appl liear ombiatios of J ad J to obtai a d solutio: def Y osπ J J si π For m iteger, Y m lim m Y still oerges. For a alue of, the geeral solutio of Bessel s DE a be writte as J Y.. Y is alled the Bessel futio of the d id. 6/68

63 Eample: 9 Sie the DE a be rewritte as 3, we hae 3. The geeral solutio is J 3 Y Y Y Bessel futios of the seod id, for,,, 3, 4 63/68

64 64/68 Eample: The Agig Sprig The DE for the free udamped motio of a mass o a agig sprig is gie b: m e αt. The hage of ariable, turs the DE ito Therefore, it s the Bessel DE with. The geeral solutio is, / t e m s α α. s ds d s ds d s. / / t t e m Y e m J t α α α α

65 Properties of Bessel Futios For m,,,, we hae: J m m J m J m m J m J m if m > ; J m, if m lim Y m Bessel futios hae the reurree relatio: J /J J. Computig high order futios from low order oes Deried from differetial reurree relatios 65/68

66 Differetial Reurree Relatio Bessel futios satisf differetial reurree relatios as follows: J J J J J J To proe the relatios, first, we hae to show that d d [ ] J J [ ] J J. ad d d The reurree relatios a be deried easil, e.g., d [ ] J J d J J J. J J 66/68

67 67/68 Differetiatio of J Assume that is a oegatie iteger ad the,! Γ J [ ]!!!!!! J d d J d d

68 Parametri Bessel s Equatio A differet form of Bessel s equatio is as follows: λ If we perform a hage of ariable, t λ, o the parametri Bessel s equatio, we will get ba the stadard Bessel s equatio: t d /dt td/dt t. Thus, the geeral solutio of the parametri form is: J λ Y λ. 68/68

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untitled ODE PDE Wat is te differece ordiar differetial eqatio partial differetial eqatio? 94. order 4. degree 7 d d d d d d 4 e e d d 4 cos 5. liear a a L a r 立數 c~6 - 7 d d d d d d 4 e e d d 4 cos 6. soltio