Microsoft PowerPoint - deq11_07_Fourier_Series_Methods.ppt

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1 Fourier Series Mehods Niol Chio Tug Uiversiy Chu-Je Tsi /4/

2 Periodic Exerl Forces Recll h lier d -order DE: d x ω x d where sds or he exerl orce imposed o he udmped sysem. Oe is periodic ucio over iervl o ieres. Quesio: Is here sysemic wy o represe geerl periodic ucio? Well Tylor series my work bu c we do beer? /33

3 Properies o Periodic Fucio Deiiio: The ucio deied or ll is sid o be periodic provided h here exiss posiive umber p such h p or ll. I p is he smlles umber wih his propery he p is clled he period o he ucio. Remrks: ier combiios o wo or more periodic ucios will sill be periodic ucio. I we use se o periodic ucios s bsis ucios o represe oher periodic ucios hey should work beer h i we use { x x x 3 } s i Tylor series. 3/33

4 Selecio o Periodic Bsis I 8 J. Fourier ssered h every ucio wih period c be represeed s lier combiio o si d s ollows: b si Relly? How bou he ucio: 3 4/33

5 Ier Produc o Fucios The ier produc o wo ucios d o iervl [ b] is he umber: b d. Two ucios d re orhogol o iervl [ b] i heir ier produc is zero: b d. 5/33

6 Fourier Series Noe h he se o rigoomeric ucios: { 3 si si si3 } is orhogol o he iervl [ ]. Fourier Series o ucio o [ ] is deied s : b si where d d d d b si d.. 6/33

7 7/33 Exmple: / Clcule d b s ollows:. d d d. si si d d d [ ]. si d b

8 Exmple: / Thus 4 si 4 si si si 5. Noe: he pril sum S N 4 N si eds o overshoo he limiig vlues o Gibbs s Pheomeo S S.5.5 x S 3 S x /33

9 9/33 Exmple: / We hve ± d d d d d

10 /33 si si d d dx d Exmple: / Noe h d b si. si 4 ±

11 Iervl o Iegrio I is impor o oe h or periodic ucio d d. Thereore he coeicies o Fourier series c be iegred over y period o such s [ ]. /33

12 /33 Fourier Series wih Period Deiiio: Fourier series c be geerlized o y ucio wih period : where si b d. si d b d

13 Covergece o Fourier Series Fourier series o piecewise smooh ucio coverges. Recll h ucio is piecewise coiuous o he iervl [ b] i here is iie umber o discoiuiies b such h. is coiuous o ech ope iervl i i.. A ech poi i o discoiuiy he limi o s pproches i rom eiher side wihi he iervl exiss d is iie. 3/33

14 Exmples: Piecewise Coiuous The ollowig ucio g is piecewise coiuous / ieger Boh g si/ d h oherwise re o piecewise coiuous o [ ]. 4/33

15 Piecewise Smooh A piecewise coiuous is sid o be piecewise smooh provided h is derivive ' is piecewise coiuous. Noe h piecewise coiuous ucio eed o be deied is isoled pois o discoiuiy. 5/33

16 Fourier Covergece Theorem Theorem: e be piecewise smooh ucio o he iervl ; h is d be coiuous excep iie umber o pois he he Fourier series o coverges o poi o coiuiy. A poi o discoiuiy he Fourier series coverges o he verge: where d deoe he limi o rom he righ d he le respecively. 6/33

17 7/33 Exmple: Coverges Discoiuiy The ollowig ucio is discoiuous x : The series coverges o. A he series coverges o:

18 Exmple: Squre Wve For he squre wve ucio 4 6 i is eve ieger he lim d lim Hece [ ]/.. This is rue sice 4 si. 8/33

19 Hl-Rge Expsios Someimes we oly cre bou he Fourier series deied o. We c deie he ucio o so h he expsio hs simpler orm. Three possible choices o exesio: y y y x x - x The hird oe hs period ohers hve period. 9/33

20 Eve d Odd Fucios A ucio is sid o be eve i d odd i. Noe h is eve while si is odd. 3 /33

21 Properies o Eve/Odd Fucios The produc o wo eve ucios is eve The produc o wo odd ucios is eve The produc o eve d odd ucios is odd The sum dierece o wo eve ucios is eve The sum dierece o wo odd ucios is odd I is eve he I is odd he d d. d. /33

22 Exmple: We c expd o he rge d mke i eve eve or odd odd ucio: d eve or odd [ ] or. eve odd Cosie series b Sie series The Fourier expsio o eve hs oly ie erms while odd hs oly sie erms. /33

23 Fourier Cosie d Sie Series Suppose h he ucio is piecewise coiuous o he iervl [ ]. The Fourier ie series o is: wih The Fourier sie series o is: b si wih b si d. d. 3/33

24 Exmple: or / For ie series d d Thus he Fourier ie series o or is 3 4 d u udu 4 [ ] or odd; u si u u or eve /33

25 Exmple: or / For sie series b si d u si udu [ u u si u]. 3 4 Thus he Fourier ie series o or is 3 si si si. 3 5/33

26 6/33 Diereiio o Fourier Series / Theorem: I is coiuous ucio wih period d is piecewise smooh or ll he he Fourier series o is he series obied by diereiio o he Fourier series Proo: is piecewise smooh hece is Fourier series coverges d we hve. si b si b

27 7/33 Diereiio o Fourier Series / where Similrly we hve si β α α [ ] d period Q α si d d α b. α. β #

28 8/33 Fourier Series Soluios o DE Assumig h we hve boudry vlue problem x" bx' cx ; x x. We c solve he problem by perorm hl-rge expsio o o d ssume h Subsiuig x x' d x" io he DE we c solve or d b. d si B A A. si b x

29 Exmple: x"4x 4 xx / Assume h or we c use odd exesio wih o ge he Fourier sie series o : 8 4 si. The soluio x should be i sie series orm s well: x b Noe h x sisies he boudry codiios. Subsiue he soluio o he DE we hve 4 b si. si 8 si. 9/33

30 Exmple: x"4x 4 xx / The soluio o he coeicies b is he b 8 x 8 4. The Fourier series soluio c be expressed s: si. 4 which is equivle o.5 x si si i he iervl /33

31 3/33 Iegrio o Fourier Series Theorem: Suppose is piecewise coiuous periodic ucio wih period d Fourier series which my o coverge. The he series is coverge or ll. Noe h i he i is o Fourier series.. si ~ b. si b ds s

32 3/33 Exmple: / The Fourier series o is Hece. si 5 5 si 3 3 si si 5 5 si 3 3 si 4 ds s s s ds s F

33 33/33 Exmple: / O he oher hd direc iegrio o yields O pge 4 o he lecure oe we kow h Thereore he erm-wise iegrio o he Fourier series o does coverge o he iegrio o.. ds s F

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