Step Functions; and Laplace Transforms of Piecewise Continuous Functions

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Doris Delphia Baker
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1 Sp Fnion; and Lapla Tranform of Piwi Conino Fnion Th prn objiv i o h Lapla ranform o olv diffrnial qaion wih piwi onino foring fnion ha i, foring fnion ha onain dioninii Bfor ha old b don, w nd o larn how o find h Lapla ranform of piwi onino fnion, and how o find hir invr ranform Or aring poin i o dy how a piwi onino fnion an b onrd ing p fnion Thn w will how h Lapla ranform and i invr inra wih h aid onr Sp Fnion Dfiniion: Th ni p fnion or Haviid fnion, i dfind by 0, <,, 0 Ofn h ni p fnion i alo dnod a, H, or H Th p old alo b mad bakward, pping down from o 0 a Thi omplmn fnion i, < 0,, Zahary S Tng C- -

2 Th Lapla ranform of h ni p fnion i L{ }, > 0, 0 Noi ha whn 0, 0 ha h am Lapla ranform a h onan fnion f Why? Thrfor, for or prpo, 0 Kp in mind ha a Lapla ranform i only dfind for 0 No: Th allaion of L{ } go a follow givn ha 0: L{ } 0 d d 0, > Zahary S Tng C- -

3 Th ni p fnion i mh mor fl han i fir appar o b Whn p in a prod wih a ond fnion, h ni p fnion a lik a wih o rn h ohr fnion on or off: 0, < f, an on wih f, f, < f, an off wih 0, By ombining wo ni p fnion, w an alo livly mak a fnion appar only for a fini draion, hn i diappar Tha i, h fnion i wihd on a a, hn i wihd off a a lar im b a b 0, f f, 0, < a a < b, b whr 0 a < b W old hink hi ombinaion a an on-off oggl wih ha onrol h apparan of h ond fnion f In ohr word, i ra a window whih w an pk ino o anohr fnion f hiding bhind Thrfor, h xprion a b i ommonly alld a window fnion, or a boxar fnion By aading h abov yp of prod, w an now wri any piwidfind fnion in a in form in rm of ni p fnion 008 Zahary S Tng C- -

4 008 Zahary S Tng C- - Sppo < < < d f b f b a f a f F n, : :,,, Thn, w an rwri F, inly, a F a f a b f b f d f n Exampl: < < 9, o 9,, F Thn, F 9 9 o

5 Th diffrn bwn f and f Exampl: π/ in and π/ in π/ Fig Graph of : π/ in Fig Graph of : π/ in π/ 008 Zahary S Tng C- -

6 Lapla ranform and ranlaion: im and frqny hif Argably h mo imporan formla for hi la, i i ally alld h Sond Tranlaion Thorm or h Sond Shif Thorm, dfining h im hif propry of h Lapla ranform: Thorm: If F L{f }, and if i any poiiv onan, hn L{ f } L{f } F No: Eqivalnly, L{ g} L{g } * Convrly, if f L {F}, hn f L { F} Exampl: Find h invr ranform of F 0 Sin F 0 L{ 0 }, hrfor, and f 0 Apply h abov horm and w hav L {F} 0 0 * Thi qivaln formla i mor xplii abo wha nd o b don whn ranforming a prod onaining a ni p fnion I ll yo o ranla h fnion,, bfor ranform h fnion Rmmbr, whn ranforming a prod onaining a p fnion: ranla bfor ranform! 008 Zahary S Tng C- - 6

7 Exampl: Find h Lapla ranform of π/ in and π/ in π/ L{ π/ in} π/ L{in π/} π/ L{o} / π Rall h addiion formla of in: in α ± β in α o β ± o α in β L{ π/ in π/} π/ F π/ L{in} / π Exampl: Find h Lapla ranform of 7 L{ 7 } L{ 7 } L{ 7 } L{ 7 } 7 7 Exampl: Find h Lapla ranform of L{ } L{ } L{ } L{ 6} Zahary S Tng C- - 7

8 Exampl: Find h Lapla ranform of, F, o, < < 9 9 W hav n arlir ha: F 9 9 o 9 o Tranform h la xprion abov, applying, whr appropria, h formla L{ g} L{g }: L{F} L{ } L{ } 9 L{o 9 9 9} L{ } L{ } 9 L{o 8 9} L{ } L{ 0 } 9 L{oo8 inin8 9} 6 9 o8 0 6 in Zahary S Tng C- - 8

9 A a paralll o h im hif propry, Lapla ranform alo ha h frqny hif propry: Thorm: If F L{f }, and if i any poiiv onan, hn L{ f } F Convrly, if f L {F}, hn f L {F } Thrfor, in h world of Lapla ranform, ranlaion ar nad by h mlipliaion wih xponnial fnion Thi horm i ally alld h Fir Tranlaion Thorm or h Fir Shif Thorm b Exampl: Ba L{o b} and L{in b} b b hn, ling a and rpla by a:, a L{ a o b} a b b a b L{ a in b} and n! Similarly, in L{ n } n, hrfor, n! L{ n a } n a 008 Zahary S Tng C- - 9

10 Diffrnial Eqaion wih Dionino Foring Fnion W ar now rady o akl linar diffrnial qaion who righ-hand id i piwi onino A mniond bfor, h mhod of Lapla ranform work h am way o olv all yp of linar qaion Thrfor, h am p n prvioly apply hr a wll Exampl: y y F, y0 0, y 0, whr F 0,, < π π Th rqir qaion i y y π Tranform h qaion and implify, w hav L{y} y0 y 0 L{y} L{ π } L{y} L{y} L{y} π π L{y} Th ond par an b invrd dirly ino in Th fir par an b invrd by fir ing aid π and hn parial fraion o implify 008 Zahary S Tng C- - 0

11 I ha a an invr ranform o Hn, h fir par i rally π L{ o } I i invrd, via h formla L{f } f, wih π, o π o π π π o Thrfor, h olion i h m of h par: y in π o in, in o, < π π No: o π o 008 Zahary S Tng C- -

12 Exampl: y 9y o π o, y0 0, y 0 0 Tranform and implify: L{y} y0 y 0 9L{y} L{o π o} L{y} 0 9L{y} 9L{y} π No: L{ π o} π L{o π} π L{o 8π} π L{o} π Thrfor, π L{y} 9 9 U parial fraion o implify h fir par: 9 9 I ha an invr ranform o o 008 Zahary S Tng C- -

13 Th ond half oni of h ngaiv of h am xprion, wih an addiional rm of π Th xra rm will ind h p fnion π, and h ranlaion ha hang ino π Hn, h ond par i invrd o π o π o π Smming p h par, h olion i, hrfor, y 0, o o o π o π o o, π < π π 008 Zahary S Tng C- -

14 Exampl: y 6y y, y0, y 0 Tranform and implify: L{y} y0 y 0 6L{y} y0 L{y} L{ } L{y} 6L{y} L{y} 6 L{y} 7 L{y} 7 Hn, 7 L{y} Th ond half i implr I an b brokn down by parial fraion ino 7 I ha an invr ranform of 008 Zahary S Tng C- -

15 008 Zahary S Tng C- - Th fir half, wiho h rm, ha parial fraion dompoiion of 0 I ha an invr ranform of 0 W hn m apply h ff of h rm, namly h inrodion of h p fnion, and h ranlaion ha hang ino Hn, hi par rally rprn 0 Combining h wo par, w now hav h olion: 0 y

16 Exampl: A ma wighing lb i aahd o a pring wih pring onan k Th ma i iniially a r in i qilibrim poiion A 0, an xrnal for of F o i applid o h ma Th for i hn abrply dionind a π Thr i no damping in h ym Find h diplamn fnion of hi ma-pring ym From h problm dripion, w dd ha: m, γ 0, k, and F π o Thrfor, h iniial val problm w nd o olv i o π o, 0 0, 0 0 Tranform h qaion and implify: L{} 0 0 L{} L{o π o} L{} 0 L{} L{} π Hn, π L{} Th fir par ha invr ranform in Th ond par, via h formla π L{f } π f π bom π π in π π π in 008 Zahary S Tng C- - 6

17 Thrfor, h olion i in π π in in, 0 < π π in, π Noi ha h ym wa ndrgoing ronan nil h foring fnion wa h off Thn i oilla a onan amplid a No: L{ ina} a Th graph of h abov olion: 008 Zahary S Tng C- - 7

18 Exri C-: Find a L{ π }, b L{ } Find a L{ π / 6 o }, b L{ π / o } Find L { } Sppo f in o π 9, find f 0, f π, f π, and f 8 6 Find ah dfini ingral by a ingraion, and b ing h propri of Lapla ranform 9 d 7 Find h Lapla ranform of, 0 < F, < 0, 6 8 Find h invr Lapla ranform of ah givn F 8 F F F F F F α β d 008 Zahary S Tng C- - 8

19 Givn ha L {oh b } b, find L { a oh b } b Givn ha L {inh b } b, find L { a inh b } 6 0 Solv ah iniial val problm 6 y 6y, y0 7 y 6y 9y, y0 0, y y y y 6, y0 0, y 0 9 y y y 0 0, y0, y y y 6, y0 0, y 0 Anwr C-: a π π π F, b F / 6 a F π π /, b F 9 8 F f 0 0, f π, f π 6, f 8 in8 o8 9 7, > Zahary S Tng C- - 9

20 008 Zahary S Tng C , > 0 7 F f f f 8 in 8 o f 6 in 6 o 8 f f β β α α α β L { a oh b } b a a L { a inh b } b a a b y y 8 in o in 6 6 y 6 in 6 o y y 0 in 0 o 6 in 7 6

The Laplace Transform

The Laplace Transform Th Lplc Trnform Dfiniion nd propri of Lplc Trnform, picwi coninuou funcion, h Lplc Trnform mhod of olving iniil vlu problm Th mhod of Lplc rnform i ym h rli on lgbr rhr hn clculu-bd mhod o olv linr diffrnil