(ebook-pdf) - mathematics - computing laplace transforms.pdf

3
&RPSXWLQJ /DSODFH 7UDQVIRUPV ( YDOXDWLQJ WKH /DSODFH , QWHJU DO 6 LQ F H WKH /DSODFH WUDQVIRUP LVGHpQHG E \DQ LQ W H J U D O LW LV F OH D U WKDW LW V H oH F W LYH FRP SXWDWLRQ ZLOO U H TX LU H VRP H S U R p F LH Q F \ LQ WKH E D V LF LQWHJUDWLRQ V NLOOV RIWKH VWDQGDUG FDOFXOXV: H ZLOO LOOX V W U D W H ZLW K D QXP EHU R I H [D P S OH V ( [DPSOH / H W I [ [ [ > 7K H Q / I V = H [ G[ ,Q W H J U D OV R I W K LV W \S H FDQ EH HYDOXDWHG X V LQ J LQWHJUDWLRQ E\ SDUWV IR U ZK LF K W K H V W D Q G D U G IR U P X OD LV = J [ K [ G[ * [ K [ b = * [ K [ G[ ZK H U H * [ G H Q R W H V D Q D Q W LG H U LYD W LYH RI J [ 7D NLQ J J [ H K [ [ D Q G X V LQ J WKH VWDQGDUG OLP LW V IR U WKH /DSODFH LQWHJUDOZH K D YH / [ V = H [ G[ b H V [ p p p V = H G[ ,W LV F OH D U IU R P W K LV WKDW WKH LQ W H J U D OLV GHpQHG IR U V! D Q G IR U VXFK YD OX H V R I V ZH K D YH / [ V V v b OLP H H w V V )R U I [ [ W K H LQWHJUDWLRQ E\SDUWV SURFHVV LV XVHG W ZLF H G LoH U H Q W LD W LQ J [ W R J H W [ D Q G W K H Q G LoH U H Q W LD W LQ J WKDW D J D LQ WR JHW ZLW K WKH UHVXOW WKDW t / [ u V V

Upload: bruce-der-schwarze

Post on 10-Nov-2015

215 views

Category:

Documents


0 download

TRANSCRIPT

  • &RPSXWLQJ /DSODFH 7UDQVIRUPV

    ( YDOXDWLQJ WKH /DSODFH , QWHJU DO 6 LQ F H W K H / D S OD F H W U D Q V IR U P LV G H p Q H G

    E \ D Q LQ W H J U D O LW LV F OH D U W K D W LW V H oH F W LYH F R P S X W D W LR Q ZLOO U H TX LU H V R P H

    S U R p F LH Q F \ LQ W K H E D V LF LQ W H J U D W LR Q V NLOOV R I W K H V W D Q G D U G F D OF X OX V : H ZLOO

    LOOX V W U D W H ZLW K D Q X P E H U R I H [D P S OH V

    ( [DPSOH / H W I [ [ [ > 7K H Q

    / I V =

    HbV[[G[

    ,Q W H J U D OV R I W K LV W \S H F D Q E H H YD OX D W H G X V LQ J LQWHJUDWLRQ E\ SDUWV IR U ZK LF K

    W K H V W D Q G D U G IR U P X OD LV

    =J [ K [ G[ * [ K [ b

    =* [ K

    [ G[

    ZK H U H * [ G H Q R W H V D Q D Q W LG H U LYD W LYH R I J [ 7D NLQ J J [ HbV[ K [ [

    D Q G X V LQ J W K H V W D Q G D U G OLP LW V IR U W K H / D S OD F H LQ W H J U D O ZH K D YH

    / [ V =

    HbV[[G[ b HbV[

    V[ppp

    V

    =

    HbV[ G[

    ,W LV F OH D U IU R P W K LV W K D W W K H LQ W H J U D O LV G H p Q H G IR U V ! D Q G IR U V X F K YD OX H V

    R I V ZH K D YH

    / [ V V

    vb OLP[

    HbV[ HbVcw

    V

    V

    )R U I [ [ W K H LQ W H J U D W LR Q E \ S D U W V S U R F H V V LV X V H G W ZLF H G LoH U H Q W LD W LQ J

    [ W R J H W [ D Q G W K H Q G LoH U H Q W LD W LQ J W K D W D J D LQ W R J H W ZLW K W K H U H V X OW W K D W

    t/ [

    uV

    V

  • ,Q J H Q H U D O Q D S S OLF D W LR Q V R I LQ W H J U D W LR Q E \ S D U W V V K R Z W K D W

    / [Q V QVQ

    + R ZH YH U W K LV LV W K H K D U G ZD \ W R R E W D LQ W K LV U H V X OW D Q H D V LH U ZD \ ZLOO E H

    G H V F U LE H G X Q G H U 3 U RSHU WLHV RI WKH /DSODFH 7 U DQVIRU P

    8VH RI &RPSOH[ ( [WHQVLRQV RI 5HDO )XQFWLRQV 7K H / D S OD F H 7U D Q V

    IR U P U H P D LQ V YD OLG IR U &RPSOH[ ( [WHQVLRQV R I W K H V W D Q G D U G U H D OYD OX H G

    H OH P H Q W D U \ IX Q F W LR Q V 8 V H R I W K LV ID F W W R J H W K H U ZLW K W K H ID F W W K D W D Q W LG LoH U

    H Q W LD W LR Q IR U P X OD V YD OLG IR U U H D O H OH P H Q W D U \ IX Q F W LR Q V U H P D LQ YD OLG IR U W K H LU

    F R P S OH [ H V W H Q V LR Q V R IW H Q D OOR ZV W K H G H V LU H G U H D O / D S OD F H LQ W H J U D O W R E H

    LG H Q W Lp H G D V W K H U H D O R U LP D J LQ D U \ S D U W R I D F R P S OH [ LQ W H J U D O ZK LF K LV P R U H

    H D V LO\ H YD OX D W H G

    ( [DPSOH 7K H / D S OD F H W U D Q V IR U P V R I F R V D[ D Q G V LQ D[ F D Q E H R E W D LQ H G

    G LU H F W O\ E \ H YD OX D W LQ J W K H / D S OD F H LQ W H J U D OV5

    HbV[ F R V D[ G[ D Q G5

    HbV[ V LQ D[ G[

    U H V S H F W LYH O\ + R ZH YH U V X F K H YD OX D W LR Q ZLW K LQ D Q H [F OX V LYH O\ U H D O Q X P E H U

    F R Q W H [W U H TX LU H V D G R X E OH D S S OLF D W LR Q R I LQ W H J U D W LR Q E \ S D U W V $ F R Q V LG H U

    D E O\ V LP S OH U U R X W H W D NH V D G YD Q W D J H R I W K H LG H Q W LW \ HL D[ F R V D[ L V LQ D[

    U H OD W LQ J W K H F R P S OH [ H [S R Q H Q W LD O IX Q F W LR Q W R W K H V H W U LJ R Q R P H W U LF IX Q F W LR Q V

    )U R P W K D W LG H Q W LW \ ZH K D YH F R V D[ 5 H HL D[ ZK H U H ? 5 H P H D Q V ? U H D O S D U W

    $ V D F R Q V H TX H Q F H ZH K D YH

    / F R V D[ V 5 H=

    HbV[ F R V D[ L V LQ D[ G[ 5 H=

    HbV[ HL D[ G[

    5 H=

    HL DbV[ G[ 5 H

    Vb LD 5 HV LD

    Vb LD V LD

    5 HV LD

    V D

    V

    V D

  • 7R R E W D LQ W K H / D S OD F H W U D Q V IR U P R I V LQ D[ ZH V LP S O\ W D NH W K H F R U U H V S R Q G LQ J

    LP D J LQ D U \ S D U W U H V X OW LQ J LQ

    / V LQ D[ V ,P V LDV D

    D

    V D

    8VH RI WKH /LQHDU LW\ RI WKH 7 U DQVIRU P )R U F R Q V W D Q W V F

    F

    D Q G IX Q F

    W LR Q V I

    [ I

    [ ZH K D YH S U R YLG LQ J D OO LQ W H J U D OV LQ G LF D W H G D U H ZH OO G H p Q H G

    / F

    I

    F

    I

    V =

    HbV[ F

    I

    [ F

    I

    [ G[

    F

    =

    HbV[ I

    [ G[ F

    =

    HbV[ I

    [ G[ F

    / I

    V F

    / I

    V

    ,Q W K LV ZD \ ZH V H H W K D W W K H OLQHDULW\ SURSHUW\ / F

    I

    F

    I

    V F

    / I

    V

    F

    / I

    V R I W K H / D S OD F H W U D Q V IR U P LV D V LP S OH F R Q V H TX H Q F H R I W K H F R U U H

    V S R Q G LQ J OLQ H D U LW \ S U R S H U W \ R I LQ W H J U D W LR Q 7K LV P D \ V H H P D E LW W U LYLD O E X W

    X V H R I W K LV S U R S H U W \ LV D Q LP S R U W D Q W W R R O LQ LG H Q W LI\LQ J F H U W D LQ / D S OD F H W U D Q V

    IR U P V

    ( [DPSOH 7K H K\SHUEROLF FRVLQH D Q G K\SHUEROLF VLQH IX Q F W LR Q V D U H G H

    p Q H G LQ W H U P V R I W K H H [S R Q H Q W LD O IX Q F W LR Q V E \ P H D Q V R I W K H OLQ H D U F R P E LQ D

    W LR Q V

    F R V K D[ HD[ HbD[

    V LQ K D[

    HD[ b HbD[

    U H V S H F W LYH O\ 6 LQ F H W K H H [S R Q H Q W LD O IX Q F W LR Q V HD[ D Q G D Q G HbD[ K D YH / D S OD F H

    W U D Q V IR U P V VbD

    D Q G VD

    U H V S H F W LYH O\ W K H OLQ H D U LW \ S U R S H U W \ V K R ZV W K D W

    / F R V K D[ V

    Vb D

    V D

    V

    V b D

    / V LQ K D[ V

    Vb D b

    V D

    D

    V b D