CHAPTER 4 LAPLACE TRANSFORM

Transcription

1 CHAPTER APACE TRANSORM

2 TOPICS DEINITION O APACE TRANSORM TABE O APACE TRANSORM PROPERTIES O APACE TRANSORM INVERSE O APACE TRANSORM PROPERTIES O INVERSE APACE TRANSORM SOVING DIERENTIA EQUATION

3 DEINITION O APACE TRANSORM aplac ranorm xinc horm a ha i i a picwi coninuou on vry ini inrval in 0, or all 0,, hn xi or all 0. Th aplac Tranorm o i dnod by and dind by: 0 d

4 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i = 0 d

5 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i = 0 d Thror,, 0

6 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i a 0 d

7 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i a 0 d Thror, a a, a

8 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i ina 0 d

9 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i ina 0 d Thror, a a in a

10 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i 0 d

11 DEINITION O APACE TRANSORM Exampl. Uing h diniion o aplac ranorm, ind i 0 d Thror,

12 DEINITION O APACE TRANSORM Exampl.5 Uing h diniion o aplac ranorm, ind or h ollowing uncion:,, 0

13 Exampl.6 Uing h diniion o aplac ranorm, ind or h ollowing uncion: DEINITION O APACE TRANSORM, 0,

14 Exampl.6 Uing h diniion o aplac ranorm, ind or h ollowing uncion: DEINITION O APACE TRANSORM, 0,

15 DEINITION O APACE TRANSORM Exampl.7 i Drmin h picwi uncion or h givn graph. ii ind h aplac ranorm o h uncion obaind in i by uing h diniion o aplac ranorm.

16 TABE O APACE TRANSORM a n a in a co a inh a coh a a n! n a a a a a a a

17 TABE O APACE TRANSORM Exampl.8 Uing h abl o aplac ranorm, ind or h ollowing uncion: a b c d 5 g h i 5 5in co inh j coh

18 TABE O APACE TRANSORM Soluion.8 a 5 5 b g 5 9 c h d i 6 j

19 PROPERTIES O APACE TRANSORM Thr ar propri o aplac ranorm will b dicud: a inar Propry o aplac Tranorm b ir Shi Propry o aplac Tranorm c aplac Tranorm o Muliplying by n uncion

20 PROPERTIES O APACE TRANSORM: Thorm. INEAR PROPERTY I and g ar xi, and i α and β ar conan, hn g g g G

21 Exampl.9 ind h aplac ranorm o h ollowing uncion: PROPERTIES O APACE TRANSORM: INEAR PROPERTY c b a co coh in

22 Soluion.9a PROPERTIES O APACE TRANSORM: INEAR PROPERTY 96! 5 5 5

23 Soluion.9b PROPERTIES O APACE TRANSORM: INEAR PROPERTY 6 coh in coh in coh in

24 Soluion.9c PROPERTIES O APACE TRANSORM: INEAR PROPERTY co co co

25 Thorm. I PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY, and a i a conan, hn a a

26 Exampl.0 ind h aplac ranorm o h ollowing uncion: PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY d c b a in coh5 co inh

27 Soluion.0a PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY!

28 Soluion.0b PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY 9 9 inh inh

29 Soluion.0c PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY 5 5! co co G g G G g g

30 Soluion.0d PROPERTIES O APACE TRANSORM: IRST SHIT PROPERTY in coh5 in coh5 G g G G g g

31 PROPERTIES O APACE TRANSORM: MUTIPYING BY n Thorm. n Givn wih n,,,..., and i, hn d d n n n n

32 Exampl. ind h aplac ranorm o h ollowing uncion: d c b a coh in co in PROPERTIES O APACE TRANSORM: MUTIPYING BY n

33 Soluion.a d d d d Sinc n =, hn ind h drivaiv o PROPERTIES O APACE TRANSORM: MUTIPYING BY n

34 Soluion.b 9 in 9 in in d d Sinc n =, hn ind h nd drivaiv o PROPERTIES O APACE TRANSORM: MUTIPYING BY n

35 d d d d d d d d - drivaiv nd drivaiv : Apply Produc Rul PROPERTIES O APACE TRANSORM: MUTIPYING BY n

36 Soluion. c co co co d d Sinc n =, hn ind h drivaiv o PROPERTIES O APACE TRANSORM: MUTIPYING BY n

37 co d d Apply Quoin Rul PROPERTIES O APACE TRANSORM: MUTIPYING BY n

38 INVERSE O APACE TRANSORM: Thorm I, hn i calld h invr aplac ranorm o and i i wrin a: -

39 INVERSE O APACE TRANSORM: Uing abl o aplac ranorm, ind h invr uncion: a b c d 5 6-0M

40 INVERSE O APACE TRANSORM: Uing abl o aplac ranorm, ind h invr uncion: a b c d co5 in inh

41 Exampl. ind h invr o aplac ranorm o h ollowing uncion: INVERSE O APACE TRANSORM: 9 5 d c b a d c b a inh in 6 5 0M

42 Anwr.a a

43 Anwr.b 6 5! 6 5!! 5!! 5, 5 5 n n! n n n

44 Anwr.c a inh,

45 Anwr.c a in 9, 9 9

46 Thorm I and xi, and i α and β ar conan, hn: - INVERSE O APACE TRANSORM: INEAR PROPERTY g G - g G G - - -

47 INVERSE O APACE TRANSORM: INEAR PROPERTY Exampl. ind h invr o aplac ranorm o h ollowing uncion: a b 5 6 a co 5 in b inh coh 5 5 0M

48 INVERSE O APACE TRANSORM: IRST SHIT O INVERSE APACE Thorm - I or can b wrin a - TRANSORM and a i a conan, hn a - a - a a

49 INVERSE O APACE TRANSORM: IRST SHIT O INVERSE APACE Exampl. TRANSORM ind h invr o aplac ranorm o h ollowing uncion: a b 5 9 a b co in 0M

50 INVERSE O APACE TRANSORM: IRST SHIT O INVERSE APACE Exampl.5 TRANSORM ind h invr o aplac ranorm o h ollowing uncion: a b 6 0 a b co in 0M

51 INVERSE O APACE TRANSORM: IRST SHIT O INVERSE APACE Exampl.6 TRANSORM ind h invr o aplac ranorm o h ollowing uncion: a b c 5 5 a b in 5 c coh inh co M

52 Exampl.7 ind h invr o aplac ranorm o h ollowing uncion: c b a 9 6 INVERSE O APACE TRANSORM: RATIONA RACTION c b a in co M

53 APPICATION O APACE TRANSORM: SOVING INITIA VAUE PROBEMS y Y y - y y 0M

54 APPICATION O APACE TRANSORM: SOVING INITIA VAUE PROBEMS aplac ranorm or dirniaion: y Y y Y y0 y Y y0 y0 y Y y0 y0 y 0 n n n n n y Y y0 y0 y 0 0M

55 APPICATION O APACE TRANSORM: SOVING INITIA VAUE PROBEMS Exampl.8 Solv h givn iniial problm y y ; y0 y 9

56 APPICATION O APACE TRANSORM: SOVING INITIA VAUE PROBEMS Exampl.9 Solv h givn iniial problm y y ; y0 0, y0 y 5 5 co 8 5 in

57 INVERSE O APACE TRANSORM: RATIONA RACTION Exampl.0 Solv h ollowing IVP: a y y in co; y0 b y y y ; y0 0, y0 a y b y 8 co 6 in