Numerical Inverse Laplace Transform Using

World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:9, 2015



1XPHULFDO,QYHUVH/DSODFH7UDQVIRUP8VLQJ &KHE\VKHY3RO\QRPLDO 9LQRG0LVKUD'LPSOH5DQL 7KHQ f t LVVDLGWREHRIH[SRQHQWLDORUGHU J  DV t o f 



International Science Index, Mathematical and Computational Sciences Vol:9, No:9, 2015 waset.org/Publication/10004329

Abstract²,Q WKLV SDSHU QXPHULFDO DSSUR[LPDWH /DSODFH

)RU J J   WKH DERYH OLPLW PD\ RU PD\ QRW EH VDWLVILHG > S@ 7KHRUHP([LVWHQFH7KHRUHP $QHFHVVDU\FRQGLWLRQIRU FRQYHUJHQFH RI LQWHJUDO GHILQHG E\   LV WKDW 5Hs ! J  

WUDQVIRUP LQYHUVLRQ DOJRULWKP EDVHG RQ &KHE\VKHY SRO\QRPLDO RI VHFRQGNLQGLVGHYHORSHGXVLQJRGGFRVLQHVHULHV7KHWHFKQLTXHKDV EHHQ WHVWHG IRU WKUHH GLIIHUHQW IXQFWLRQV WR ZRUN HIILFLHQWO\ 7KH LOOXVWUDWLRQVVKRZWKDWWKHQHZGHYHORSHGQXPHULFDOLQYHUVH/DSODFH WUDQVIRUPLVYHU\PXFKFORVHWRWKHFODVVLFDODQDO\WLFLQYHUVH/DSODFH WUDQVIRUP

ZKHUHFRQVWDQW J  LVH[SRQHQWLDORUGHURI f t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

 Keywords²&KHE\VKHY SRO\QRPLDO 1XPHULFDO LQYHUVH /DSODFH WUDQVIRUP2GGFRVLQHVHULHV , ,1752'8&7,21 A. The Laplace Transform +(WKHRUHWLFDOIRXQGDWLRQWRDFODVVRILQWHJUDOWUDQVIRUPV LV GXH WR)UHQFK PDWKHPDWLFLDQ 3LHUUH 6LPRQ GH /DSODFH   ZKR PDGH XVH RI LQWHJUDO LQ KLV ZRUN RQ SUREDELOLW\WKHRU\>S@>S@/DSODFHWUDQVIRUP LV WKH PRVW SRSXODU LQWHJUDO WUDQVIRUP LQ ZKLFK D IXQFWLRQ f t  RI RQH YDULDEOH t  WUDQVIRUPV LQWR IXQFWLRQ F s  RI DQRWKHU GRPDLQ s V PD\ EH FRPSOH[  0DWKHPDWLFDOO\ WKH /DSODFH WUDQVIRUP RI D IXQFWLRQ f t  GHQRWHG E\ L> f t @  LV GHILQHGDV

7



F s

f

L> f t @

³e

 st

f t dt      



  st

SURYLGHG LQWHJUDO H[LVWV e LV FDOOHG WKH NHUQHO RI WKH WUDQVIRUPDWLRQ>S@ 'HILQLWLRQ  $ IXQFWLRQ f t  LV VDLG WR EH SLHFHZLVH

B. Inverse Laplace Transform ,I L> f t @ F  s  WKHQ WKH LQYHUVH /DSODFH WUDQVIRUP RI IXQFWLRQ f t ZKLFKPDSVWKH/DSODFHWUDQVIRUPRIDIXQFWLRQ EDFNWRWKHRULJLQDOIXQFWLRQLVGHILQHGDV

VHFWLRQDO  FRQWLQXRXV RQ WKH LQWHUYDO >a b@  LI LW LV ERXQGHG DQGKDVDWPRVWILQLWHO\PDQ\GLVFRQWLQXLWLHVLQWKDWLQWHUYDO 'HILQLWLRQ  /HW f t EH SLHFHZLVH FRQWLQXRXV IXQFWLRQ



L  F  s

GHILQHGIRUDOOSRVLWLYH t LQWKHUDQJH  f ZLWKWKHSURSHUW\

$QDO\WLFLQYHUVLRQRI/DSODFHWUDQVIRUPLVLQIDFWUHODWHGWR WKH WKHRU\ RI FRPSOH[ YDULDEOHV 7KH FODVVLFDO DQG VLPSOHVW LQYHUVLRQ IRUPXOD LV JHQHUDOO\ FDOOHG %URPZLFK LQWHJUDO RU %URPZLFK 0HOOLQ &RQWRXU LQWHJUDO RU VRPHWLPHV +HDYLVLGH LQYHUVLRQIRUPXOD>@>@ 7KHRUHP  7KH %URPZLFK ,QYHUVLRQ 7KHRUHP  >@ /HW f t SRVVHVVDFRQWLQXRXVGHULYDWLYHDQG f t  ke Jt  ZKHUH

WKDWWKHUHH[LVWVDUHDOQXPEHU J  VXFKWKDW 

 

OLP f t e Jt

t of

­ J ! J  ® ¯r f J  J 

f t  







k DQG J DUH WKH SRVLWLYH FRQVWDQWV 'HILQH   7KHQ LQYHUVLRQ

9LQRG 0LVKUD LV ZLWK WKH 'HSDUWPHQW RI 0DWKHPDWLFV 6DQW /RQJRZDO ,QVWLWXWH RI (QJLQHHULQJ DQG 7HFKQRORJ\ /RQJRZDO 3XQMDE ,QGLD FRUUHVSRQGLQJ DXWKRU SKRQH  HPDLO YLQRGPLVKUD#UHGLIIPDLOFRP  'LPSOH 5DQL LV ZLWK WKH 'HSDUWPHQW RI 0DWKHPDWLFV 6DQW /RQJRZDO ,QVWLWXWH RI (QJLQHHULQJ DQG 7HFKQRORJ\ /RQJRZDO 3XQMDE ,QGLD HPDLO GLPSOHFKDZOD#\DKRRFRP 

RI F s LVJLYHQE\WKHLQWHJUDO 



International Scholarly and Scientific Research & Innovation 9(9) 2015

581

scholar.waset.org/1999.7/10004329

World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:9, 2015



f t

 Si

c  if

³ F s e

st

7KH LQWHUYDO  ’  WUDQVIRUPHG LQWR  S  DQG f t 

ds 

EHFRPHV

c  if



5H s

§  · f ¨  ORJ VLQ T ¸ © V ¹

c LV VR FKRVHQ WKDW WKH SROHV VLQJXODULWLHV  OLH WR

WKH OHIW RQ WKH OLQH 5Hs

c  8VXDOO\ F s LV DQ DQDO\WLF



International Science Index, Mathematical and Computational Sciences Vol:9, No:9, 2015 waset.org/Publication/10004329

IXQFWLRQ RQ WKH KDOI SODQH 5Hs ! J   F s KDV VRPH



C.Numerical Inverse Laplace Transform 6RPHWLPHV LW LV YHU\ DUGXRXV RU HYHQ LPSRVVLEOH WR LQYHUW LQWRRULJLQDOWLPHGRPDLQIXQFWLRQIURPLWVIUHTXHQF\GRPDLQ FKDUDFWHUL]DWLRQ GXH WR GLIILFXOW\ LQ ILQGLQJ WKH SROHV DQG UHVLGXHV RI F s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x 0HWKRGVEDVHGRQ*DXVVLDQTXDGUDWXUH x 0HWKRGVZKLFKH[SDQG f t LQH[SRQHQWLDOIXQFWLRQ x 0HWKRGVEDVHGRQ)RXULHUVHULHV x 0HWKRGVEDVHGRQUDWLRQDODSSUR[LPDWLRQ /HJHQGUH IXQFWLRQ *DXVV /HJHQGUH TXDGUDWXUH UXOH 3DGH DSSUR[LPDQW /DJXHUUH VHULHV DQG )RXULHU VHULHV PHWKRG DUH FHUWDLQ PHWKRGV IHDVLEOH LQ OLWHUDWXUH >@ >@ >@ >@ IRU QXPHULFDOO\LQYHUWLQJWKH/DSODFHWUDQVIRUP,Q>@6KHQJHW DO KDYH VXFFHVVIXOO\ WHVWHG ,QYODS *DYVWHK DQG QXPHULFDO LQYHUVH /DSODFH WUDQVIRUP 1,/7  DOJRULWKPV IRU ILQGLQJ LQYHUVH /DSODFH WUDQVIRUPV IRU IUDFWLRQDO RUGHU GLIIHUHQWLDO HTXDWLRQV



S s

³ VLQ T V



FRV T g T dT   





%\VHWWLQJ s k   V  k «ZHKDYH S 

³ VLQ T

V F k   V





k

FRV T g T dT    



+HUH ZH DVVXPH WKDW gS f    ,Q FDVH WKLV LV QRW VDWLVILHGWKHQDUUDQJHLWE\VXEWUDFWLQJDVXLWDEOHIXQFWLRQIURP g T 7KHIXQFWLRQ g T FDQEHH[SDQGHGLQ  S DVWKHRGG FRVLQHVHULHV  g T

f

¦D

k

 

FRV k   T

  

k 



DQG LV YDOLG LQ WKH LQWHUYDO  S  S   7KXV ZLWK k

 ! ZH

JHW  

 S     

V F V D   



S

        

 V F V D   D  

 

S



  k   k V F k   V

­° kª§k · §k·º ½° rª§k · §k ·º ® «¨¨ ¸¸ ¨¨ ¸¸»D! «¨¨ ¸¸ ¨¨ ¸¸»Dkr !Dk ¾  °¯ °¿ ¬«©k ¹ ©k ¹¼» ¬«©r ¹ ©r ¹¼»



7KXV D k FDQEHREWDLQHGIURP E\WKHIRUZDUGVXEVWLWXWLRQ DQGKHQFH g T FDQEHREWDLQHGIURP  'HILQH U k   x FRV kT  x VLQ T 

ZKHUH

FRV T

U k x 

WKH

&KHE\VKHYSRO\QRPLDORIVHFRQGNLQGRIGHJUHHNLV7KHUHE\ XVLQJ FRV T   e Vt     ZHH[SUHVV  f  x



+HUH ZH IROORZ SDUDOOHO WHFKQLTXH RI 3DSRXOLV >@ DV LQWURGXFHGLQ>@ZKHUHE\PDNLQJWKHVXEVWLWXWLRQ





 V F  s

,,180(5,&$/,19(56,21%$6('21&+(%@



a

%\ HPSOR\LQJ WKH PHWKRG ILUVWO\ ZH FKHFN WKH FRQGLWLRQ f   E\WKHLQLWLDOYDOXHWKHRUHP,QWKLVFDVH

 

 OLP sF  s

OLP s

s of

s of

s sa

International Science Index, Mathematical and Computational Sciences Vol:9, No:9, 2015 waset.org/Publication/10004329









 I W

f 



$V LW LV VDWLVI\LQJ WKH DVVXPSWLRQ VR WKH WHFKQLTXH LV DSSOLFDEOHKHUH7KXVWKHVROXWLRQ f t FDQEHREWDLQHG 7DEOH , UHSUHVHQWV WKH FRHIILFLHQWV REWDLQHG IURP UHODWLRQ  ,Q7DEOH,,ZHKDYHFRPSDUHGWKHQXPHULFDODSSUR[LPDWH VROXWLRQZLWKH[DFWVROXWLRQRILQYHUVH/DSODFHWUDQVIRUP

   

 7$%/(,

 

&$/&8/$7,212)&2()),&,(176,17+((;3$16,212) f t  N

Dk 

          

          

N                   

a  e t

s

 ZLWK WKH

>@)LUVWO\



OLP sF  s

OLP se  a

s of

s

s

OLP

s of

e

a s



 W













D

k                    

 7$%/(,9 &203$5,62162)35(6(17$3352;,0$7(,19(56(:,7+(;$&7/$3/$&( ,19(56( W ([DFWVROXWLRQ 1XPHULFDOLQYHUVH $EVROXWHHUURU    (    (    (    (    (    (    (    (    (    (

ZHZLOOFKHFNWKHFRQGLWLRQXVLQJ/¶+RSLWDO¶VUXOH

s of



&$/&8/$7,212)&2()),&,(176,17+((;3$16,21

([DPSOH  &RQVLGHU WKH IXQFWLRQ F  s e  a  St 



 7$%/(,,,



a



)LJ&RPSDULVRQRI([DFWDQG$SSUR[LPDWH,QYHUVH/DSODFH 7UDQVIRUPLQ([DPSOH

 7$%/(,, &203$5,62162)(;$&762/87,21$1'35(6(17$3352;,0$7(62/87,21 W ([DFWVROXWLRQ 3UHVHQWDSSUR[LPDWHVROXWLRQ $EVROXWHHUURU   ( (    ( (       ( (    (       ( (       ( (       (

H[DFWLQYHUVH/DSODFHWUDQVIRUPDV f t

([DFWVROXWLRQ 3UHVHQW$SSUR[LPDWHVROXWLRQ





+HQFH WKH JLYHQ IXQFWLRQ LV VDWLVI\LQJ WKH DVVXPSWLRQ 7KXV WKH1XPHULFDO/DSODFHWUDQVIRUPRIJLYHQIXQFWLRQLVIRXQG 

International Scholarly and Scientific Research & Innovation 9(9) 2015

583

scholar.waset.org/1999.7/10004329

World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:9, 2015

 

7KH JLYHQ IXQFWLRQ LV VDWLVI\LQJ WKH UHVWULFWLRQ 7KHUHIRUH WKH1XPHULFDO/DSODFHWUDQVIRUPRIDERYHIXQFWLRQLVREWDLQHG DQGLVSUHVHQWHGWKURXJK)LJE\FRQVLGHULQJ a  





,9 &21&/86,21 7KH QXPHULFDO DSSUR[LPDWH VROXWLRQV LQ H[DPSOHV H[KLELW WKH DSSOLFDELOLW\ RI WHFKQLTXH WR ILQG WKH LQYHUVH /DSODFH WUDQVIRUP RI IXQFWLRQV 6ROXWLRQ H[SUHVVHG LQ WHUPV RI RGG FRVLQH VHULHV LV EDVHG RQ &KHE\VKHY SRO\QRPLDO 7KH UHVXOW VKRZVWKDWWKHPHWKRGLVHIILFLHQWDQGHDV\WRLPSOHPHQW

I W





5()(5(1&(6 >@ $ 0 &RKHQ Numerical Methods for Laplace Transform Inversion 6SULQJHU >@ 5(%HOOPDQ++.DJLZDGDDQG5(.DOED³1XPHULFDO,QYHUVLRQ RI /DSODFH 7UDQVIRUPV DQG 6RPH ,QYHUVH 3UREOHPV LQ 5DGLDWLYH 7UDQVIHU´Journal of Atmospheric Sciences, YROSS >@ /'¶$PRUH*/DFFHWWLDQG$0XUOL³$Q,PSOHPHQWDWLRQRID)RXULHU 6HULHV 0HWKRG IRU WKH 1XPHULFDO ,QYHUVLRQ RI WKH /DSODFH 7UDQVIRUP´ $&07UDQVDFWLRQVRQ0DWKHPDWLFDO6RIWZDUHYRO1RSS±  >@ %'DYLHVDQG%0DUWLQ³1XPHULFDO,QYHUVLRQRI/DSODFH7UDQVIRUP$ 6XUYH\ DQG &RPSDULVRQ RI 0HWKRGV´ Journal of Computational PhysicsYROSS >@ $ 3DSRXOLV ³$ 1HZ 0HWKRG RI ,QYHUVLRQ RI WKH /DSODFH 7UDQVIRUP´ Quart. Appl. Math.,YROSS >@ 9 0LVKUD ³5HYLHZ RI 1XPHULFDO ,QYHUVH RI /DSODFH WUDQVIRUPV XVLQJ )RXULHU$QDO\VLV )DVW)RXULHUWUDQVIRUPDQG2UWKRJRQDOSRO\QRPLDOV´ Mathematics in Engineering Science and Aerospace, YRO  SS   >@ -/6FKLIIThe Laplace Transform: Theory and Applications6SULQJHU  >@ )$8ULEH-/1DUHGR30RUHQRDQG/*XDUGDGR³(OHFWURPDJQHWLF 7UDQVLHQWV LQ 8QGHUJURXQG 7UDQVPLVVLRQ 6\VWHPV WKURXJK WKH 1XPHULFDO /DSODFH 7UDQVIRUP´ International Journal of Electrical Power and Energy Systems, YROSS >@ 9 0LVKUD DQG ' 5DQL ³&KHE\VKHY 3RO\QRPLDO %DVHG 1XPHULFDO ,QYHUVH /DSODFH 7UDQVIRUP 6ROXWLRQV RI /LQHDU 9ROWHUUD ,QWHJUDO DQG ,QWHJURGLIIHUHQWLDO (TXDWLRQV´ American Research Journal of MathematicsYROSS >@ 5 ( %HOOPDQ 5( .DODED DQG -$ /RFNHWW Numerical Inversions Laplace Transforms with Applications $PHULFDQ (OVHYLHU 3XEOLVKLQJ &R1HZ@ - *O\Q Advanced Modern Engineering Mathematics 3HDUVRQ (GXFDWLRQ,QGLDQUHSULQW  >@ + 6KHQJ @ -/:X&+&KHQDQG&)&KHQ³1XPHULFDO,QYHUVLRQRI/DSODFH 7UDQVIRUP XVLQJ +DDU :DYHOHW 2SHUDWLRQDO 0DWULFHV´ IEEE Transactions on Circuits and Systems-I: Fundamental Theory and ApplicationsYROSS    'U 9LQRG 0LVKUD KROGV 3URIHVVRU SRVLWLRQ LQ 0DWKHPDWLFV DW 6/,(7 /RQJRZDO ,QGLD  +H ZDV )HOORZ DW ,QGLDQ ,QVWLWXWH RI $GYDQFHG 6WXG\ 6KLPOD6RIDUKHKDVVXSHUYLVHGWZRUHVHDUFKVFKRODUVIRU3K'  0V'LPSOH5DQLLVFXUUHQWO\DUHVHDUFKVFKRODUDW6/,(7/RQJRZDO,QGLD  SXUVXLQJ3K'XQGHU'U9LQRG0LVKUD  

([DFW/DSODFHWUDQVIRUP 1XPHULFDO,QYHUVH/DSODFHWUDQVIRUP  



















W



)LJ&RPSDULVRQRI([DFWDQG$SSUR[LPDWH,QYHUVH/DSODFH 7UDQVIRUPLQ([DPSOH  



 I W

International Science Index, Mathematical and Computational Sciences Vol:9, No:9, 2015 waset.org/Publication/10004329





 ([DFW,QYHUVHWUDQVIRUP 1XPHULFDO,QYHUVH/DSODFHWUDQVIRUP  





















W

)LJ&RPSDULVRQRI([DFWDQG$SSUR[LPDWH,QYHUVH/DSODFH 7UDQVIRUPLQ([DPSOH 

([DPSOH&RQVLGHUWKHIXQFWLRQ  F s

e a

s



s



ZLWKWKHH[DFWLQYHUVH/DSODFHWUDQVIRUPJLYHQE\  f t

e

a t

St

>@



7R VDWLVI\ WKH FRQGLWLRQ RI WKH DIRUHPHQWLRQHG WHFKQLTXH E\ XVLQJLQLWLDOYDOXHWKHRUHPDQGXVLQJ/¶+RSLWDOUXOHZHDWWDLQ  OLP sF  s

s of

OLP se  a

s of

s

OLP

s of

s ea

s



 

International Scholarly and Scientific Research & Innovation 9(9) 2015

584

scholar.waset.org/1999.7/10004329