Second Order Linear Partial Differential Equations. Part II

Transcription

1 Secod Order iear Parial Differeial Equaios Par II Fourier series; Euler-Fourier formulas; Fourier Covergece Theorem; Eve ad odd fucios; Cosie ad Sie Series Eesios; Paricular soluio of he hea coducio equaio Fourier Series Suppose f is a periodic fucio wih a period T. The he Fourier series represeaio of f is a rigoomeric series ha is i is a ifiie series iss of sie ad ie erms of he form a f a b si Where he coefficies are give by he Euler-Fourier formulas: a m m f d m 3 b f si d 3 The coefficies a s are called he Fourier ie coefficies icludig a he a erm which is i realiy he -h ie erm ad b s are called he Fourier sie coefficies. 8 Zachary S Tseg E- -

2 Noe : Thus every periodic fucio ca be decomposed io a sum of oe or more ie ad/or sie erms of seleced frequecies deermied solely by ha of he origial fucio. Coversely by superimposig ies ad/ or sies of a cerai seleced se of frequecies we ca reruc ay periodic fucio. Noe : If f is piecewise coiuous he he defiie iegrals i he Euler- Fourier formulas always eis i.e. eve i he cases where hey are improper iegrals he iegrals will coverge. O he oher had f eeds o o be piecewise coiuous o have a Fourier series. I jus eeds o be periodic. However if f is o piecewise coiuous he here is o guaraee ha we could fid is Fourier coefficies because some of he iegrals used o compue hem could be improper iegrals which are diverge. Noe 3: Eve hough ha he sig is usually used o equae a periodic fucio ad is Fourier series we eed o be a lile careful. The fucio f ad is Fourier series represeaio are oly equal o each oher if ad wheever f is coiuous. Hece if f is coiuous for < < he f is eacly equal o is Fourier series; bu if f is piecewise coiuous he i disagrees wih is Fourier series a every discoiuiy. See he Fourier Covergece Theorem below for wha happes o he Fourier series a a discoiuiy of f. Noe : Recall ha a fucio f is said o be periodic if here eiss a posiive umber T such ha f T f for all i is domai. I such a case he umber T is called a period of f. A period is o uique sice if f T f he f T f ad f 3T f ad so o. Tha is every ieger-muliple of a period is agai aoher period. The smalles such T is called he fudameal period of he give fucio f. A special case is he a fucios. Every a fucio is clearly a periodic fucio wih a arbirary period. I however has o fudameal period because is period ca be a arbirarily small real umber. The Fourier series represeaio defied above is uique for each fucio wih a fied period T. However sice a periodic fucio has ifiiely may ofudameal periods i ca have may differe Fourier series by usig differe values of i he defiiio above. The differece however is really i a echical sese. Afer simplificaio hey would look he same. 8 Zachary S Tseg E- -

3 Therefore echically a leas a Fourier series of a periodic fucio depeds boh o he fucio as well as is chose period. Noe : The defiie iegrals i he Euler-Fourier formulas ca be foud be iegraig over ay ierval of legh. However from o is he coveio ad is ofe he mos coveie ierval o use. Noe 6: Sice he Fourier coefficies are calculaed by defiie iegrals which are isesiive o he value of he fucio a fiiely may pois. Cosequely piecewise coiuous fucios of he same period ha differ from each oher a fiiely may pois oably a isolaed discoiuiies per period will have he same Fourier series. Noe 7: The a erm i he Fourier series which has epressio a f d f d is jus he average or mea value of f o he ierval [ ]. Sice f is periodic his average value is he same for every period of f. Therefore he a erm i a Fourier series represes he average value of he fucio f over is eire domai. 8 Zachary S Tseg E- - 3

4 8 Zachary S Tseg E- - Eample: Fid a Fourier series for f < < f f. Firs oe ha T hece. The a erm is oe half of: d d m f a The res of he ie coefficies for 3 are d d f a si si si d Hece here is o ozero ie coefficie for his fucio. Tha is is Fourier series coais o ie erms a all. We shall see he sigificace of his fac a lile laer.

5 8 Zachary S Tseg E- - The sie coefficies for 3 are si si d d f b si d eve odd. Therefore si f.

6 Figure: he graph of he parial sum of he firs 3 erms of he Fourier series f si. Compare i agais he graph of he acual fucio he series represes he fucio f < < f f see earlier. 8 Zachary S Tseg E- - 6

7 8 Zachary S Tseg E- - 7 Eample: Fid a Fourier series for f < < f f. How will i be differe from he series above? 8 d a For 3 : d a si si d b 8 si Cosequely si si b a a f.

8 Eample: Fid a Fourier series for f < < f f. Aswer : 8 f Eample: Fid a Fourier series for < f f f. < 8 Aswer : f si 8 Zachary S Tseg E- - 8

9 Comme: Jus because a Fourier series could have ifiiely may ozero erms does o mea ha i will always have ha may erms. If a periodic fucio f ca be epressed by fiiely may erms ormally foud i a Fourier series he he epressio mus be he Fourier series of f. This is aalogous o he fac ha he Maclauri series of ay polyomial fucio is jus he polyomial iself which is a sum of fiiely may powers of. Eample: The Fourier series period represeig f si is jus f si. Eample: The Fourier series period represeig f 6 si is o eacly iself as give sice he produc si is o a erm i a Fourier series represeaio. However we ca use he double-agle formula of sie o obai he resul: 6 si 3 si. Cosequely he Fourier series is f 3 si. 8 Zachary S Tseg E- - 9

10 The Fourier Covergece Theorem Here is a heorem ha saes a sufficie codiio for he covergece of a give Fourier series. I also ells us o wha value does he Fourier series coverge o a each poi o he real lie. Theorem: Suppose f ad f are piecewise coiuous o he ierval. Furher suppose ha f is defied elsewhere so ha i is periodic wih period. The f has a Fourier series as saed previously whose coefficies are give by he Euler-Fourier formulas. The Fourier series coverge o f a all pois where f is coiuous ad o lim f lim f / c c a every poi c where f is discoiuous. Comme: As see before he fac ha f is piecewise coiuous guaraees ha he Fourier coefficies ca be foud. The codiio ha f is also piecewise coiuous is a sufficie codiio o guaraee ha he series husly foud will be coverge everywhere o he real lie. As well recall ha suppose f is coiuous a c he by defiiio f c equals boh oesided limis of f as approaches c. Therefore he secod par of he heorem could be eve more succicly saed as ha he Fourier series represeig f will always coverge o lim f lim f / c c a every poi c ad o jus a discoiuiies of f. A equece of his heorem is ha he Fourier series of f will fill i ay removable discoiuiy he origial fucio migh have. A Fourier series will o have ay removable-ype discoiuiy. 8 Zachary S Tseg E- -

11 Eample: e us revisi he earlier calculaio of he Fourier series represeig f < < f f. The Fourier series as we have foud is f si. The followig figures are he graphs of various fiie -h parial sums of he series above. 3 8 Zachary S Tseg E- -

12 3 8 Zachary S Tseg E- -

13 Noe ha superimposed siusoidal curves ake o he geeral shape of he piecewise coiuous periodic fucio f almos immediaely. As well for he pars of he curve where f is coiuous where he Fourier Covergece Theorem predics a perfec mach he composie curve of he Fourier series coverges rapidly o ha of f as prediced. The covergece is o as rapidly ear he jump discoiuiies. Ideed for all bu he lowes parial sums of he Fourier series he curve seems o overshoo ha of f ear each jump discoiuiy by a oiceable margi. Furher more his discrepacy does o fade away for ay fiiely larger. Tha is he covergece of a Fourier series while predicable is o uiform. Tha is a small price we pay for approimaig a piecewise coiuous periodic fucio by siusoidal curves. I ca be doe bu he Fourier series does o coverge uiformly o he acual fucio. This behavior is kow as he Gibbs Pheomeo. I furher saes ha he parial sums of a Fourier series will overshoo a jump discoiuiy by a amou approimaely equal o 9% of he jump. Tha is ear each jump discoiuiy he overshoo amous o abou.9 lim f lim f c c for large. Furher his overshoo does o go away for ay fiiely large. 8 Zachary S Tseg E- - 3

14 Quesio: Skech he graph of he Fourier series of f < < f f. We have see a few graphs of is parial sums. Bu wha will he graph of he acual Fourier series look like? Eample: Skech he graph of he Fourier series of f < < f f. Eample: Skech he graph of he Fourier series of < f f f. < 8 Zachary S Tseg E- -

15 Eve ad Odd Fucios Recall ha a eve fucio is ay fucio f such ha f f for all i is domai. Eamples: sec ay a fucio 6 A odd fucio is ay fucio f such ha f f for all i is domai. Eamples: si a csc co 3 3 Mos fucios however are eiher eve or odd. There is oe fucio ha is boh eve ad odd. Wha is i? Arihmeic Combiaios of Eve ad Odd Fucios The able below summaries he resul of performig he commo arihmeic operaios o a pair of eve ad/or odd fucios: Eve ad Eve Odd ad Odd Eve ad Odd / Eve Odd Neiher / Eve Eve Odd The resul above ca be eeded o arbirarily may erms. For eample a sum of hree or more eve fucios will agai be eve. Care eeds o be ake i he cases where 3 or more odd fucios formig a produc/quoie. For eample a produc of 3 odd fucios will be odd bu a produc of odd fucios is eve. 8 Zachary S Tseg E- -

16 Calculus Properies of Eve ad Odd Fucios Suppose f is a eve fucio coiuous o he f d f d. Suppose f is a odd fucio coiuous o he f d. 8 Zachary S Tseg E- - 6

17 The Fourier Cosie Series Suppose f is a eve periodic fucio of period he is Fourier series coais oly ie iclude possibly he a erm erms. I will o have ay sie erm. Tha is is Fourier series is of he form a f a. Coversely ay periodic fucio whose Fourier series has he form of a ie series as show mus be a eve periodic fucio. Compuaioally his meas ha he Fourier coefficies of a eve periodic fucio are give by a m m m f d f d m 3 b 3 Noice ha he iegrad i he defiie iegral used o fid he ie coefficies a s is a eve fucio i is a produc of wo eve fucios f ad. Therefore we ca use he symmeric propery of eve fucios o simplify he iegral. 8 Zachary S Tseg E- - 7

18 The Fourier Sie Series If f is a odd periodic fucio of period he is Fourier series coais oly sie erms. I will o have ay ie erm. Tha is is Fourier series is of he form f b si Coversely ay periodic fucio whose Fourier series has he form of a sie series as show mus be a odd periodic fucio. Therefore he Fourier coefficies of a odd periodic fucio are give by a m m 3 b f si d 3 Eample: We have calculaed earlier ha he fucio f < < f f has as is Fourier series iss of purely sie erms: f si. We ow see ha his sie series sigifies ha he fucio is odd periodic. I is perhaps o very obvious bu he iegrad i he iegral for he Fourier sie coefficies is aoher eve fucio. I is a produc of wo odd fucios f ad si which makes i eve. Therefore we ca agai ake advaage of he symmeric propery of eve fucios o simplify he iegral. 8 Zachary S Tseg E- - 8

19 The Cosie ad Sie Series Eesios If f ad f are piecewise coiuous fucios defied o he ierval he f ca be eeded io a eve periodic fucio F of period such ha f F o he ierval [ ] ad whose Fourier series is herefore a ie series. Similarly f ca be eeded io a odd periodic fucio of period such ha f F o he ierval ad whose Fourier series is herefore a sie series. The process ha such eesios are obaied is ofe called ie /sie series half-rage epasios. Here is a oulie of how his ca be doe. Sar wih a fucio ha is defied oly o a ierval of fiie legh from o. Firs epad he fucio o be defied o he ierval from o such ha he fucio is a eve or a odd fucio as required. The defie he fucio o be periodic wih a period of T by requirig F F. This process is acually much easier ha i souds. Mahemaically he process ca be achieved raher simply as described below. Eve ie series eesio of f Give f defied o [ ]. Is eve eesio of period is: f F F F. f < < a Where F a such ha a m m f d m 3 b 3 8 Zachary S Tseg E- - 9

20 Odd sie series eesio of f Give f defied o. Is odd eesio of period is: f F f < < F F. < < Where F b a m si such ha m 3 b f si d 3 Eample: e f eesios of period. <. Fid is ie ad sie series Aswers: Cosie series: 8 f Sie series: f si 8 Zachary S Tseg E- -

21 8 Zachary S Tseg E- - Back o he Hea Coducio Problem Previously we had foud he geeral soluio of he iiial-boudary value problem give by he oe-dimesioal hea coducio equaio modelig a bar ha has boh of is eds kep a degree. The geeral soluio is e C u α si /. Seig ad applyig he iiial codiio u f we ge si f C u. We ow kow ha he above equaio says ha he iiial codiio eeds o be a odd periodic fucio of period. Sice he iiial codiio could be a arbirary fucio i usually meas ha we would eed o force he issue ad epad i io a odd periodic fucio of period. Tha is si b f. Compare he wo epressios we see ha si si b f C u. Therefore he paricular soluio is foud by seig all he coefficies C b where b s are he Fourier sie coefficies of or he odd periodic eesio of he iiial codiio f : d f b C si.

22 Eample: Solve he hea coducio problem 8 u u < < > u ad u u si si si. Sice he sadard form of he hea coducio equaio is α u u we see ha α 8; ad we also oe ha. Therefore he geeral soluio is u C C e e α / 8 / si si The iiial codiio f is already a odd periodic fucio oice ha i is a Fourier sie series of he correc period T. Therefore o addiioal calculaio is eeded ad all we eed o do is o erac he correc Fourier sie coefficies from f. To wi C b C b C b C b for all oher or. Hece u e e 8 8 / / si si e 8 / si 8 Zachary S Tseg E- -

23 8 Zachary S Tseg E- - 3 Wha will he paricular soluio be if he iiial codiio is u isead? Tha is solve he followig hea coducio problem: 8 u u < < > u ad u u. The geeral soluio is sill / 8 si e C u. The iiial codiio is a odd fucio bu i is o a periodic fucio. Therefore i eeds o be epaded io is odd periodic eesio of period T. Is coefficies are for 3 si si d d f b si d eve odd

24 The resulig sie series is represeig he fucio f < < f f : f si. The paricular soluio ca he be foud by seig each coefficie C o be he correspodig Fourier sie coefficie of he series above C b. Therefore he paricular soluio is u e 8 / si. 8 Zachary S Tseg E- -

25 Eercises E-.: 8 Fid he Fourier series represeaio of each periodic fucio. Deermie he values o which each series coverge o a.. f < < f f.. f si < < f f. 3. f 6 3 < f f.. f < f f.. f < < f f. < < 6. f < < f f. 7. f si < < f f. 8. f δ c < < < c < f f. 9 Epad each fucio io is ie series ad sie series represeaios of he idicaed period. Deermie he values o which each series coverge o a ad. 9. f 3 T 6.. f e T. <. f si T.. f T 6. < 3 3. Solve he hea coducio problem u u < < 9 > u ad u9 u si / 3 si / 3 si3.. Solve he hea coducio problem of he give iiial codiios. 9 u u < < > u ad u a u 3si si7 / 6 6si b u c u. 8 Zachary S Tseg E- -

26 Aswers E-.:. f f.. f si f f 3 si f f si. f f. 3 f. 6. f si si f. 7. f f. 8. f c f. 9. Cosie series: 3 f 3 Sie series: 9 f si ; 3 The ie series coverges o 3 ad ; he sie series coverges o ad respecively a ad.. Cosie series: f e e Sie series: f si ; The ie series coverges o e a all 3 pois. The sie series coverges o e ad e respecively a ad.. Cosie series: f Sie series: f si ; Boh series coverge o a ad o si a. A he ie series coverges o si he sie series o si. 8 Zachary S Tseg E- - 6

27 8 Zachary S Tseg E Cosie series: f Sie series: 3 si 3 si 6 f ; Boh series coverge o a ad o a. A he ie series coverges o he sie series o. 3. si3 3 si 3 si 8 /9 3 /9 e e e u. a si si si 3 36 / 9 9 e e e u ; b / 9 si 6 e u.