Symbolic Determination of Laplace Transform

8/12/2019 Symbolic Determination of Laplace Transform

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A L G O R I T H M S F O R S Y M B O LIC D E T E R M I N A T I O NO F L A P L A C E T R A N S F O R M * )

A. MIOLAIs t i tu to M atem atico dell Un iversit~ di Ro m a

1. I N T R O D U T I O NThis paper descr ibes a recursive method for the symbol ic determinat ion of

the Laplace t ransform, d i rect or inverse, of a funct ion belonging to a wel l def inedc l a s s

D u r i n g t h e I F I P w o r k i n g co n f e ren ce o n s y m b o l man i p u la t io n lan g uag es, h e ldin P isa i n Sep tembe r 1966 , P ro f . Carl Eng lem an and o ther r esearch wo rker s f romthe MITRE-Corpora t ion o f Bedfo rd p resen ted a very in t e res t i ng paper wh ich ,among o ther t h ings , cons ider s t he p rob lem of the Lap lace t r ans fo rm [ 3 ] [4 ] .Tak ing in to cons idera t ion some o f t he r esu l t s ob t a ined by Eng leman wepresen t a m etho d which en l arges r emarkab ly the f ami ly o f func t ions t o be man i -pulated thus al lowing to meet numerous appl icat ions of the analysis .

2. D E S C R I P T IO N O F T H E A L G O R I T H M2.1 Family o[ t rans formable [unc t ions .

Th e fam i ly f l; of t ransform able funct ions F t ) has been rec ursively def inedas fo l lows: t he fo l lowing func t ions, ca l l ed base o r e l emen tary func t ions) be longto fl;1 k2) (a t + b)3) e t

*) W ork carried out in the fram e of the Italian CNR Research Grou p No. 43.852


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2.2 T r a n s / o r m a t i o n m e t h o d .Th e p r o ced u r e f o r d e te r mi n in g t h e tr an s f o r m ( an t i t r an s f o r m) o f a F ( t ) ( / ( s ) )

takes p lace in three s tages . In the f i rs t s tage some subst i tu t ions are made on themen t ioned func t ion , poss ib ly r educing func t ions s ine and cos ine t r i gonomet r i c o rhype rbol ic , expo nent ial s on w hatev er basis in to expone nt ial s on basis e , B esselfunct ions of the second kind in to Bessel funct ions of the f i rs t k ind.

Th erefo re i f t he cons idered func t ion comes ou t f rom the app l ica t ion o f oneof t he opera to r s D I, D2, D3, D4, Ds , D6 DI , D4, D6, D7 t o ther func t ions , t heprob lem reduces t o t he t r ans fo rmat ions o f t he l a t t e r , mak ing use o f one o f t hek n o w n p r o p e r ti e s o f t h e o p e r at o r L ( L - 1 ) ( s ee [ 1 ] ) .

I f on the con t ra ry the cons idered func t ion i s no t o f t he t ype descr ibed , wepass t o t he t h i rd s t age in which i f i t comes ou t t o be an e l em en tary func t ion i t st r ans fo rm i s g iven ; o therwi se i t i s exp la ined why the t r ans fo rm has no t been done .R e m a r k s :

1) Th e an t i tr ans fo rms o f t he r a t i ona l func t ions a re d e t e rm ined by thefo l lowing fo rmula :

[ / s > ] = ~k=l h=l i n k - - h ) ~ ( h - - l ) /

wh ere g ( s ) = f i (S- -~k)m*~=

/ (s ) o f o rd er < m r + m 2 + . . . + m ncP -* I [ S ) S --a R )m *~ s ) - - d sh -1 g s )

2) .The show m m ethod does no t keep in to accoun t t he ex i s t ence o f t het rans fo rm, o f wh ich one mus t be su re befo rehand .

3. D E S C R I P T I O N O F T H E P R O G R A M S3.1 L A P T R a n d I N L T

Th e m e t h o d p r e s en t ed in t h e p reced in g p ar ag rap h s w as p r o g r am med i n L I S P ,in a f i r s t ver s ion fo r t he IBM 7090 o f t he C .N.U.C .E. o f P i sa , and in t he p resen to n e f o r t h e C I N A C o f t h e I N A C o f R o me .

Th e L I S P - f u n c ti o n s LA P T R an d I N LT o p e r a ti n g r e s pec t iv e ly t h e d i r ec t an dinver se t r ans fo rm, have th ree a rgument s : t he express ion to be t r ans fo rmed , i t s854

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p r i n c ip a l v a r i a b l e a n d t h e p r i n c ip a l v a r i a b l e ( w h i c h c a n b e t h e r e f o r e c h o s e n a tw i l l ) o f t h e e x p r e s s i o n o b t a i n e d a s r e s u l t .

T h e t h r e e s t a g es o f t h e p r o c e d u r e ( s e e 2 . 2 ) f o r d e t e rm i n i n g t h e t r a n s f o r m( a n ti tr a n sf o r m ) r e q u e s t e d t o L A P T R ( I N L T ) a r e c a r ri e d o u t : t h e f ir st o n e f r o mt h e L I S P - fu n c t io n s F U N T R A a n d C O M M , t h e s ec o n d o n e f ro m L A P 1 ( I N L 1 ) ,a n d t h e t h ir d o n e ~r om L A P 2 ( I N L 2 ) . T h e s e a b o v e m e n t i o n e d L I S P - f u n c t io n sh a v e o n l y o n e a rg u m e n t : t h e e x p r e s s i o n o n w h i c h t h e y o p e r a t e .

N o t e t h a t I N L 2 a n t i tr a n s f o r m s t h e r a ti o n a l f u n c t i o n s a c c o r d in g t o th e a lg o -r i th m s h o w n ( s ee 2 . 2 ).

T h e L I S P - f u n c t i o n S I N A C w i t h z e r o a r g u m e n t s , i n it ia t e s th e t ra n s f o r m a t i o np r o c e d u r e , c al li n g f o r t h e o p e r a t i o n t o b e c a rr i e d o u t ( L A P T R o r I N L T ) ; t h e n ,a f t e r r e c e iv i n g t h e i n f o r m a t io n , c a ll s f o r th e a r g u m e n t s o n w h i c h t o o p e r a t e : t h ee x p r e s s i o n t o b e t r a n s f o r m e d ( w h i c h i s t r a n s l a te d f r o m i ts e x t e r n a l i n f ix e d f o r mt o i ts in t e r n a l p re f i x e d f o r m , f r o m t h e in p u t L I S P - f u n c t i o n L I S P I N ) , a n d th ep r i n c i p a l v a r i a b l e s ; f i n a ll y a f t e r t h e t r a n s f o r m a t i o n i t p r i n t s t h e r e s u l t a l r e a d yr e p o r t e d i n t h e e x te r n a l fo r m w i t h t h e L I S P O U T f u n c ti o n .

T h e i n t e r n a l r e p r e s e n t a t i o n o f t h e e x p r e s s i o n s i s t h e o n e i n p r e f i x e d n o t a t i o nw i t h a ll th e b r a c k e t s n e c e s sa r y t o L I S P ; t h e e x t e rn a l o n e is d e t e r m i n e d a s f o l l o w si n B a c k u s n o r m a l f o r m :

< f i g u r e > : : = 0 1 1 1 2 [ 3 1 4 1 5 1 6 1 7 1 8 1 9character> : : A I B I C { D I E I F I G I H I l I J I K I L I M i N [ O { P I Q I R [ S I T I U I V I W I X I Y I Z

< - ) - o p > : : - - 1 /< + o p > : : -- + [ --< nu mb er > : : = < figure [ < nu m be r > < figure >< at om > : : --- < character > [ < a to m > < character > I < at om > < nu mb er >< e x p r e s s i o n > : : = < n u m b e r > I < a t o m > [ ( < s u m > ) l ( < f u n c t i o n > ) : : = < e x p r e s s i o n > [ 1 ' < e x p r e s s i o n > : : = ] < - ) ( - o p > < s u m > : : = [ < s u m > < + o p > [ - < , f u n c t i o n > : : = < a t o m > < l i st a rg s >< l i s t a r g s > : : = < e x p r e s s i o n > I < l i s t a r g s > < e x p r e s s i o n >

T h e i n p u t l a n g u a g e j u s t d e f i n e d , h o w e v e r , m u s t b e r e s t r i c t e d c o m p a t i b l y w i t ht h e e x p r es s io n s o f t h e f u c ti o n s re c o g n i z ed b y L A P T R a n d I N L T .

S o m e p a rt ic u l a r c o n s t a n t s h a v e a n a t o m f i x e d f o r t h e ir r e p r e s e n t a t i o n s , t h a ti s : E f o r t h e b a s i s o f t h e e x p o n e n t i a l s , P I f o r z : , E C f o r t h e E u l e r ' s c o n s t a n t

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y ; t he r ecogn ized func t ions a re on the con t ra ry SQRT , SIN, COS, SIN H , C OSH ,T A N G , L N , E R F C , O N E F , G A M M A , J B E S; m o r e o v e r S E R I E S K m e a n s ~ a n dFA LT UN G deno tes t he i n teg ra l p roduct , k=0

3.2 Subprograms.A m o n g t h e s u b p r o g rams u s ed b y LA P T R an d I N L T, s o me o n es h av e p ar ti cu la rimpor t ance .T h e p r o g r a m D / D which carr ies out the symbol ic der ivat ion of algebraic

funct ions, t r igonometr ic d i rect and inverse fuct ions, exponent ial and logar i thmicfunc t ions and o f t he func t ion o f t he e r ro r s Er f .

Th e p r o g r ams S - k , S - - , S * , S / S ~ a re ca l led in b y any o the r p rogram whichbui lds up algebraic expressions, they redu ce the expressions them selves consol idat -ing the numer i ca l t e rms and car ry ing ou t symbol i ca l ly t he opera t ions shown in t heexpressions according to the rules of the l i teral calculus.

4 . C O N C L U S I O N SKeep ing in to accoun t t he func t ions which may be long to t he f ami ly 3 ; (~ -~) ,

no te t ha t t he g iven method a l lows to meet numerous p rob lems o f t he numer i ca lanalysis a nd calculus; th is is in cont raposi t ion wi th the E nglem an s wo rk w hich,opera t ing on ra t i ona l , exponen t i a l and t r i gonomet r i c func t ions was par t i cu l a r ly ap tfo r dea l ing w i t h a lgebra ic p rob lems .

I t m us t a l so be no ted tha t t he en l a rgem ent o f t he f am i ly ~ ; (~ - 1 ) impl i edthe modi f i ca t ion o f t he a lgor i t hm and therefo re a d i f f e ren t i dea o f t he p rograms ,for a wider general i ty . I t i s in fact possib le to enlarge fur ther on the fami ly~(~ ; -1 ) o f t r ans fo rmab le func t ions i n t roducing o ther e l emen tary func t ions , o ro ther op era to r s , wi thou t m od i fy ing the a lgor it hm.

5. O N E A P P L I C A T I O NN o w w e s ee h o w s o me ex amp le s o f t h o se ca r ri ed o u t b y mean s o f LA P T R

and INLT, can be used fo r a p rob lem refer r ing to t he hea t t r ansmiss ion in so l idbodies (see [ 5] ) .

Le t us cons ider two bar s concur ring in a node , and assume tha t i n t h e nodebe loca t ed a source o f hea t ; mo reover l e t u s assume tha t t h e l oss o f hea t i n t hemedium be negl ig ib le .

I. f t he bar s a re exac t ly a like , t he h ea t em i t t ed by the source wi ll be sharedequal ly be tw een the two bars . I f on the co n t ra ry the t herm ic p roper t i es o f t hebar s a re d i f f e ren t , t he d i s t r i bu t ion l aw wi l l vary accord ing to t he t ime, even i f856


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i t i s cond i t ioned by the equal ity of temp era ture in the sec t ions converging tothe node .

Then having ass igned the law q ( t ) of the hea t emit ted by the source , we haveto compute the l aws q . t ) and qa ( t ) o f the hea t r ece ived by the two ba rs .

Moreove r the t empera tu re s o f the two ba rs in the node mus t be equa l .We reca l l here the law express ing the tempera ture wi th respec t to the f lux inthe nodal sec t ion:

1 a fo t e-ta


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h a v i n g p u tK = F12 b l 2 - - F22 b2 2F12 -- F22

w i t h Flbl :~ F2 b2 and F , :~ F2 we ob ta inq i p ) F1 2 p + b l 2

qo F12 - - F 2 2 p p + K )Ft F2 p + b t a ) p - + . b 2 2 )

F t 2 - - F 2 2 p p + K )

V P q - b l 2 ) ( P ] - b 22)and ana logous ly fo r q 2 p )q oW e cons ider t he on ly q l t ) , q2 t ) and we shal l obtain by di f ference.H a v i n g p u t

F12 /012A 1 - - 9F12 F 2 2 K

A 2 F 1 2 K - - b l 2F t 2 - - F 22 KEl W2.B t F I 2 - - F 2 2

.B2 F1 F2 b l 2 b 22F I 2 - - F 22 K

B 3 =

w e h a v e :

F 1 F 2 ( K - b l 2) ( K - b 2 2)F t 2 F 2 2 K

q l p ) A I A 2 [ B 2= ~ + B t + ~ +q o p p + K k pB 3 1

p + K ~[ p + b , 2 ) p+ b2 )A n t i t r a n s f o r m i n g t h is l a s t e x p r e s s io n u s i n g I N L T , , s ee e x a m p l e s ) , w e g e t :

q l t ) - - A t + A z e - k t - - B 1 / t ) - - 132 / ( z ) d z - -q

~ tB 3 e - B e k / ( ~ ) d z8 8


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w h e r ebit + b t/ ( t ) = e- I ~ r b22) t ) q z ( t) - ql(t)qo

6 . A P P E N D I XIn order to obtain a t ransform (ant i t ransform) i t suff ices to in i t ia l ize the pro-

ced u r e b y mean s o f S I N A C an d t o su p p ly o n r eq u est : LA P TR ( I N L T) an d th e i ra rgument s ( see examples ) .

Th e causes o f a non-execu tion o f t he r eq ues t ed t r ans fo rm, po in t ed ou t by aner ro r wi th p rogress ive numera t ion can be :

1 ) Der iva t ion o f a func t ion non recogn ized by D/D.2 ) Transfo rm at ion o f t he t ype L [ t k (F ( t ) ] w i th k non in t eger.3 ) Transfo rm at ion o f t he t ype Fl( t )F2( t) .4) Transfo rmat ion o f t he t ype e a~ F ( t ) w i t h n > l .5 ) T ransfo rma t ion o f a func t ion non recogn ized by LA PT R.6) T ransfo rma t ion o f a ~unction non recogn ized by INL T.

Examples :( S I N A C )L A P T R( (O N EF T) - - 2 (ON EF (T - - A) ) + (ON EF (T - - 2 3(- A) ) )TS( 1 / S - - 2 --X-E ~' (- - A % S ) / S + E ~ ( - 2 - ) 6 A % S ) / S )L A P T R(E 1' ( -- B 1' 2 3(-- T ) / (S Q RT T ) )TS(S Q RT (P I / (S + B 1' 2) ) )I N L T(1 / (S (S -- A) 1 ' 2) )ST1 / A t 2 + T % E ' ~ ( A % T ) / A - - E ~ ( A % T ) / A ~ 2 )I N L T(1 / (S Q RT ( (S + A ) -)(- (S + B) ) ) )ST(E '~ ( - - (A + B ) - ) 6 T / 2 ) - ) 6 ( IB E S ( ( A - - B) T / 2) O) )

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R E ~ R E N ~[1] GHIZZET rI, A.: Calcolo simbolico. Zanichelli, Bologna 1943.[2] DOETSCH,G.: Intro duc tion , i l util ization pratiqu e de la tras/ormation de L aplace. Gauthier-V i l l a r s Paris 1959.[3] ENGLEMAN,C.: M AT H LA B : a program /or on-l ine machine ass is tence in symbol ic comp u-tations. M ITR E- Corporat ion MT P 18, ot tobre 1965.[4 ] ENGLEm~a~, C.: R a t io n a l / u n c t io n s i n MA THLA B . MITRE-Corpo ra t i on MTP 35, ago-sto 1966.[ 5] T u ~ , M .: Stato termico e dilatazione degli elementi prismatici . I.P.S.I. - Pompei 1965.

D i s c u s s i o n i

A . A S C A R I : C h i e d o a l p r e s e n t a t o r e c o n f e r m a d e l c a r a t t e r e d i de l suo con t r i bu to , o s se rva ndo ch e , i n u na ce r t a c l a s se d i app l i ca -z i o n i , ~ p a r t i c o l a r m e n t e i n t e r e s s a n t e l a p o s si b il it ~ d i o t t e n e r e t r a s f o r m a t e i n v e r s ein fo rm a ana l i t i c a , spec i e c on l u so de l l op e ra to r e

Symbolic Determination of Laplace Transform

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