drexel institute of technology
TRANSCRIPT
. I
NASA CX-72412
D I T Report 160-10
THE LAPLACE TRANSFORM
OF THE DERIVATIVE OF A FUNCTION WITH FINITE JUMPS
by
Leon Y. Bahar
Prepared for
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Lewis Research Center
JUNE 1968
NASA Contract NsG-270
DREXEL INSTITUTE OF TECHNOLOGY
NOTICE
This r e p o r t w a s prepared as a n account of Government sponsored work. Neither t h e United States, nor t h e Na t iona l Aeronaut ics and Space Adminis t ra t ion (NASA), nor any pe r son a c t i n g on behalf of NASA:
A.) Makes any warranty o r r e p r e s e n t a t i o n , expressed o r implied, with r e s p e c t t o t h e accu racy , completeness, o r u s e f u l n e s s of t h e information con ta ined i n t h i s r e p o r t , o r t h a t t h e use of any in fo rma t ion , appa ra tus , method, o r p rocess d i s c l o s e d i n t h i s r e p o r t may not i n f r i n g e p r i v a t e l y owned r i g h t s ; o r
B . ) Assumes any l i a b i l i t i e s w i th r e s p e c t t o t h e use o f , o r f o r damages r e s u l t i n g from t h e u s e of any i n f o r - mation, appa ra tus , method o r p rocess d i s c l o s e d i n t h i s r e p o r t .
A s used above, “person a c t i n g on behalf of NASA” i n c l u d e s any employee o r c o n t r a c t o r of NASA, o r employee of such con- t r a c t o r , t o t h e e x t e n t t h a t such employee o r c o n t r a c t o r of NASA, o r employee of such c o n t r a c t o r p repa res , d i s s e m i n a t e s , or provides a c c e s s t o , any information pu r suan t t o h i s employment o r c o n t r a c t w i th NASA, o r h i s employment w i t h such c o n t r a c t o r .
Requests f o r c o p i e s of t h i s r e p o r t should be r e f e r r e d t o :
Nat ional Aeronaut ics and Space Admin i s t r a t ion O f f i c e of S c i e n t i f i c and T e c h n i c a l Information A t t e n t i o n : AFSS-4 Washington, D.C. 20546
NASA CR-72412
DREXEL Report 160-10
THE LAPLACE TRANSFORM
OF THE DERIVATIVE OF A FUNCTION WITH FINITE JUMPS
by
Leon Y. Bahar
Prepared f o r
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
L e w i s Research Center
JUNE 1968
Contract NsG-270
Technical Management NASA L e w i s Research Center
Chemical Rocket D iv i s ion Cleveland, Ohio
Richard N . Johnson
Drexel I n s t i t u t e of Technology 32nd and Chestnut S t r e e t s
Ph i l ade lph ia , Pennsylvania 19104
?
SUMMARY
General ized func t ions as developed by Laurent Schwartz are used t o
inc lude f i n i t e jump c o n t r i b u t i o n s t o t ransforms of d e r i v a t i v e s . The
expres s ion of t h e B i l a t e r a l Laplace Transform of t h e d e r i v a t i v e of a
p iecewise continuous func t ion i s f i r s t der ived and then s p e c i a l i z e d t o
t h e u n i l a t e r a l transform.
Transform of a genera l ized d e r i v a t i v e r e s o l v e s some well-known apparent
c o n t r a d i c t i o n s .
and "ac tua l" i n i t i a l va lues d i sappea r s .
i s r e s t r i c t e d t o o rd ina ry d i f f e r e n t i a l equa t ions , i t can be extended t o
p a r t i a l d i f f e r e n t i a l equat ions .
It i s shown t h a t t h e use of t h e Laplace
I n t h i s new con tex t t h e d i s t i n c t i o n between "prescr ibed"
Although t h e p re sen t i n v e s t i g a t i o n
,
I-
I -
@ .
i
. I
I . INTRODUCTION
I n t h e a n a l y t i c a l s tudy of meteoroid impact on space v e h i c l e s , i nc lud ing
f u e l tanks , t h e i n t e g r a l t ransform technique i s one of t h e a v a i l a b l e methods.
The impact loadings , whether on the bumper o r on t h e main s t r u c t u r a l w a l l ,
are always a b r u p t l y appl ied .
represented by d iscont inuous func t ions .
methods, t h e t rea tment of discont inuous f u n c t i o n s e x h i b i t i n g f i n i t e jumps
as g iven i n most textbooks and papers i s less than s a t i s f a c t o r y . I n t h i s
paper , a more adequate t rea tment u t i l i z i n g t h e n o t i o n of genera l ized f u n c t i o n s
i s given.
These abrupt l oad ings may be mathematical ly
I n t h e a p p l i c a t i o n of t ransform
The d e r i v a t i v e of a func t ion wi th f i n i t e d i s c o n t i n u i t i e s cannot be
transformed i n accordance wi th the w e l l known express ion:
L { f ' ( t > ) = s F ( s ) - f(O+) (1-1)
which only a p p l i e s t o continuous func t ions . The c o n t r i b u t i o n of f i n i t e jumps
t o t h e r igh thand s i d e of t h i s equat ion i s u s u a l l y g iven as a n e x e r c i s e of no
p a r t i c u l a r s i g n i f i c a n c e i n most textbooks (1,2). I n phys ica l a p p l i c a t i o n s ,
such as hyperve loc i ty impac t problems, piecewise cont inuous func t ions are of
g r e a t importance, and t h e i r behavior w i l l u sua l ly c o n t a i n a t least one jump
occur r ing a t zero. The expression (1-1) does no t i nc lude t h e c o n t r i b u t i o n
due t o t h i s term, since t h e t ransform i s def ined as t h e l i m i t of t h e Laplace
I n t e g r a l as t h e lower l i m i t tends t o zero from t h e r i g h t . To inc lude such
e f f e c t s t h e range of i n t e g r a t i o n can be extended t o minus i n f i n i t y , t hus
in t roduc ing t h e Bi la teral Laplace Transform ( 3 ) . This procedure has l i m i t e d
use fu lness , s i n c e i t can only be used t o t ransform a known funct ion . I n t h e
s o l u t i o n of d i f f e r e n t i a l equations one has, i n gene ra l , no advance information
r*
i
,
about t h e l o c a t i o n and magnitude of d i s c o n t i n u i t i e s t h a t w i l l appear i n t h e
response as they must be p a r t of t h e s o l u t i o n . Therefore , an ex tens ion of
t h e no t ion o f func t ion becomes necessary i n o rde r t o t ransform such ab rup t
changes of unknown l o c a t i o n and magnitude i n t h e response.
Generalized f u n c t i o n s as in t roduced by Laurent Schwarz ( 4 ) a r e i d e a l l y
s u i t e d t o handle t h e p a r t i c u l a r problem under cons ide ra t ion . The expres s ion
(1-1) becomes upon such a g e n e r a l i z a t i o n :
where t h e jumps a re i m p l i c i t l y conta ined i n t h e d e f i n i t i o n of t h e gene ra l i zed
func t ion .
The express ion g iven i n (1-2) can be e a s i l y extended t o t h e d e r i v a t i v e
of o rde r n:
(1-3) n- 1 (n-1) ( o - ) L { g ( " ) ( t ) j = sn L ( g ( t > ) - s g ( o - ) - .....-g
The equat ion (1-3) i s both s impler and more g e n e r a l t han i t s corresponding
expres s ion f o r an o rd ina ry func t ion , as w i l l be e s t a b l i s h e d i n l a t e r s e c t i o n s .
A c a r e f u l d i s t i n c t i o n between o rd ina ry and gene ra l i zed f u n c t i o n s can
a l s o r e s o l v e some apparent c o n t r a d i c t i o n s i n p a r t i a l d i f f e r e n t i a l equa t ions
such as those r e c e n t l y s t u d i e d by Boley (5) and Reid ( 6 ) . It i s expected
t h a t t h e i n v e s t i g a t i o n of t h i s problem w i l l be t h e s u b j e c t of another s tudy .
I n o r d e r to emphasize t h e underlying i d e a s , w e l l known s u f f i c i e n t
cond i t ions governing equa t ion (1-1) w i l l no t be given. These cond i t ions are
s t a t e d w i t h p r e c i s i o n and c o r r e c t p e r s p e c t i v e i n LePage ( 7 ) .
- 2-
I1 BACKGROUND AND PRELIMINARIES
A f u n c t i o n is s a i d t o have a f i n i t e d i s c o n t i n u i t y a t to i f f(t:) and f ( t i )
+ + e x i s t , and f ( t o ) # f ( t i ) . The q u a n t i t y J { f ( t o ) } = f ( t o ) - f ( t i ) is known as
t h e ''jump" or"sa1tus" of t h e func t ion a t t h e p o i n t to, and may b e r ep resen ted
by J { f ( t o ) } , J ( t o ) o r Jo. It can a l s o b e considered as a f u n c t i o n J ( t ) of t h e
cont inuous v a r i a b l e t , which i s zero except a t a p o i n t of d i s c o n t i n u i t y .
Th i s i s i n essence t h e "Saltus-function" whose p r o p e r t i e s have been s t u d i e d
by Hobson (5).
b e r ep resen ted by an upward arrow i f J {f ( t ) } > o and a downward arrow i f
J { f ( t o > } c 0.
g r a p h i c a l d i f f e r e n t i a t i o n of t he jump m u l t i p l i e d by t h e u n i t s t e p func t ion .
The jump can be considered as a d i r e c t e d q u a n t i t y and may
0
This r e p r e s e n t a t i o n is p a r t i c u l a r l y u s e f u l i n t h e formal
The only paper d i s c u s s i n g the i n c l u s i o n of jumps i n t h e Laplace t ransform
of t h e d e r i v a t i v e of a f u n c t i o n seems t o be t h a t of Rasof ( 9 ) .
Rasof (9) has inco rpora t ed t h e e f f e c t of jumps i n f ( t ) i n t o t h e
e x p r e s s i o n f o r L { f ' ( t ) ) , by consider ing t h e d e f i n i t i o n s
L { f ( t > } = L { f l ( t ) } + lwt1 [ f 2 ( t ) - f ~ ( t ) l e - ~ ~ d t (2-1 1
d t (2-2) -st L {f( t )} = L { f 2 ( t ) } + 1'1 [ f l ( t > - f 2 ( t > 1 e
0
where f l ( t ) and € 2 ( t ) have been def ined i n t h e open i n t e r v a l s (0, t l ) and
( t 1 , m ) as shown i n F igu re 1.
The paper makes t h e i n c o r r e c t assumption t h a t t h e n o t a t i o n L { f ( t ) ) i s
ambiguous, because i t does n o t s t a t e which of t h e two forms (2-1) o r (2-2)
must be a s c r i b e d t o f ( t ) i n i n t e r p r e t i n g L ( f ( t ) } .
exists, and t h e forms (2-1) and (2-2) i n t roduce t h e a d d i t i o n a l complication of
making i t impossible t o determine t h e Laplace t ransforms from t a b l e s .
Ac tna l ly , no such d i f f i c u l t y
- 3-
Figure 1
It i s not d i f f i c u l t t o see how complicated t h e expressions given i n
. . . . , f n } de f in ing t h e (9) would become as t h e set of func t ions { f , piecewise continuous f u n c t i o n f i nc reases .
r e p r e s e n t a t i o n s f o r L {f ( t ) } , wi th t h e ensuing complicated expres s ions
f o r L { f ' ( t ) ) t o b e found i n t h e same paper ( 9 ) .
1 f 2 9
This would l e a d t o n d i f f e r e n t
The complication i n t h e proposed problem does no t ar ise from t h e
source i n d i c a t e d by Rasof (9) .
c a t i o n of t h e i n i t i a l v a l u e s i n t h e t ransform of t h e d e r i v a t i v e of a gene ra l i zed
It i s u s u a l l y due t o a n i n c o r r e c t i d e n t i f i -
funct ion.
It should aga in b e po in ted ou t t h a t i n a g iven p h y s i c a l problem, if
t h e f u n c t i o n f ( t ) i s t h e f o r c i n g f u n c t i o n i t i s n o t d i f f i c u l t t o o b t a i n t h e
transform of i t s d e r i v a t i v e s , even i f f ( t ) is no t continuous.
some e x t e n t i s the problem t r e a t e d i n (9) .
d i f f i c u l t t o i d e n t i f y t h e l o c a t i o n and magnitude of t h e jumps t h a t appear
i n t h e response. This paper w i l l be mainly devoted t o a fo rmula t ion t h a t
Th i s , t o
It i s f a r more important and
w i l l au tomat i ca l ly accomplish t h i s a i m .
-4-
- .
I11 ELEMENTARY FORMULATION AND SOLUTION OF THE PROBLEM
Consider the function represented in Figure 2. If this function is
considered an ordinary function it has the following representation in
i: the open intervals I
f(t) = fi(t) t E Ii (i = 1, 2 , 3, 4 )
where the following definitions are used:
- m < t < t l t E I1
t 1 < t < o t E I2
t 2 < t < a t E I,
o < t < t 2 t E I3
The same function can also be represented in the form oL a regular generalized
function, by introducing the unit step function, so as to produce the functions
f.(t), t E I in the various intervals of definition. This function may be 1 i denoted by g(t) and written explicitly as:
where the use of the gate function defined as the difference of two step
functions has been made in order to annihilate the function outside the gate.
Figure 2
-5-
It i s w e l l known t h a t wh i l e t h e r e i s no reason t o d i s t i n g u i s h between t h e
two d i f f e r e n t r e p r e s e n t a t i o n s of t h e same func t ion , namely t h e o rd ina ry
f u n c t i o n f ( t ) and t h e gene ra l i zed f u n c t i o n g ( t ) ; t h e s e two f u n c t i o n s e x h i b i t
marked d i f f e r e n c e s under d i f f e r e n t i a t i o n , due t o t h e i n t r o d u c t i o n of t h e
"Dirac d e l t a " as t h e d e r i v a t i v e of t h e u n i t s t e p func t ion , The o rd ina ry
d e r i v a t i v e of f ( t ) a t a p o i n t of f i n i t e d i s c o n t i n u i t y such as t L , 0, and t 2
is undefined; t h e gene ra l i zed d e r i v a t i v e s a t those p o i n t s are g iven by
I n o rde r t o d e r i v e t h e expres s ion of t h e gene ra l i zed d e r i v a t i v e g ' ( t )
i n terms of t h e o rd ina ry d e r i v a t i v e f ' ( t ) and t h e c o n t r i b u t i o n of t h e
f i n i t e jumps, one can formal ly d i f f e r e n t i a t e t h e expres s ion f o r g ( t ) as
g iven by ( 3 - 2 ) .
I n o rde r t o achieve t h e d e s i r e d r e s u l t , i t i s more convenient t o
rewrite t h e express ion f o r g ( t ) i n a form t h a t w i l l e x h i b i t t h e jumps,
namely
g ( t > = f l ( t > + [ f 2 ( t > - f , ( t > l U(t'tl) + [ f , ( t ) - f 2 ( t ) ] u ( t )
The i n t e r p r e t a t i o n of t h i s equat ion is of i n t e r e s t i n i t s e l f . It simply
i n d i c a t e s t h a t t h e f u n c t i o n g ( t ) can be cons t ruc t ed by a s t epwise c o n t i n u a t i o n
procedure s t a r t i n g w i t h t h e f u n c t i o n f ( t ) .
[ f 2 ( t ) - f l ( t ) ] u ( t - t l ) which b r i n g s t h e f u n c t i o n up f ( t ) from t l t o i n f i n i t y ,
t h e next term b u i l d s t h e f u n c t i o n up t o f ( t ) from zero t o i n f i n i t y , and so on.
To f l ( t ) i s added 1
2
3
-6-
The formal a p p l i c a t i o n of the d i f f e r e n t i a t i o n of a product t o (3-3)
y i e l d s :
( 3 - 4 )
It i s clear t h a t t h e terms m u l t i p l i e d by t h e u n i t s t e p f u n c t i o n i n t h e
expres s ion ( 3 - 4 ) r e p r e s e n t t h e o rd ina ry d e r i v a t i v e f ' ( t ) and t h e terms
invo lv ing t h e Dirac d e l t a can be r e w r i t t e n i n a form t h a t w i l l produce t h e
same i n t e g r a l according t o t h e s i f t i n g property.
It i s clear t h a t t h e c o e f f i c i e n t s of t h e d e l t a f u n c t i o n s r e p r e s e n t t h e
jumps of t h e f u n c t i o n a t t h e p o i n t s of d i s c o n t i n u i t y . This y i e l d s :
g ' ( t ) = f ' ( t ) + J ( f ( t 1 ) ) 6 ( t - t l ) + J ( f ( 0 ) ) 6 ( t ) (3-6)
There i s no lo s s of g e n e r a l i t y i n t h i s d e r i v a t i o n and i t was presented i n
d e t a i l i n conformity w i t h F igu re 2. Summation s i g n s a t t h i s s t a g e would
have served no u s e f u l purpose i n f u r t h e r c l a r i f i c a t i o n of t h e d e r i v a t i o n .
G e n e r a l i z a t i o n now gives: n
i- 1 g ' ( t ) f ' ( t ) + C J { f ( t i ) } 6 ( t - t i )
Although t h e above d e r i v a t i o n was formal , i t can b e shown t h a t i t i s i n
agreement w i t h t h a t obtained v i a t h e Theory of D i s t r i b u t i o n s (10) and
f i n d s i t s r i g o r o u s j u s t i f i c a t i o n only i n such an i n t e r p r e t a t i o n .
- 7-
(3-7)
It would b e of i n t e r e s t t o determine t h e b i l a t e r a l t ransform of t h e
g e n e r a l i z e d d e r i v a t i v e of g ' ( t ) a f t e r determining i t f o r t h e o rd ina ry d e r - r a t i v e
Following Figure 2, t h i s expres s ion can b e w r i t t e n as: - -
0 t 2 -
L 2 { f ' ( t ) } = 1:; f ' l ( t ) e - s t d t + It+ f ' 2 ( t ) e s t d t + lo+ f ' 3 ( t ) ; s td t 1
+ 1" + f l , ( t ) e s t d t t 2
I n t e g r a t i o n by p a r t s , g i v e s under t h e u s u a l assumptions found i n (7)
- [ f (O+) - f (033 - [f ( t 2+) - f (t 33 e s t 2 3 2 4 3 2
The q u a n t i t i e s w i t h i n t h e b r a c k e t s are t h e terms i n d i c a t i n g jump
c o n t r i b u t i o n s , and t h e r e s u l t can b e e a s i l y g e n e r a l i z e d to :
- L I
L {f '<t)} = s L ( f ( t ) ) - C J { f ( t i ) } e s t i is1 2 2
The b i l a t e r a l t ransform of t h e g e n e r a l i z e d d e r i v a t i v e (3-7) g i v e s
on t h e o t h e r hand:
n
i=1 L, { g ' ( t > ) = L2 { f ' ( t > ) + C J { f ( t i ) } e s t i
Combining t h e two expres s ions (3-10) and (3-11) y i e l d s
(3-8)
(3-10)
(3-11)
(3-12)
C l e a r l y L
gene ra l i zed f u n c t i o n con ta in ing only u n i t s t e p f u n c t i o n s i n i t s d e s c r i p t i o n .
(f ( t ) ) = L { g ( t ) } s i n c e t h e f u n c t i o n g ( t ) i s a r e g u l a r 2 2
Theref o r e
(3-13)
The s i m p l i c i t y of t h i s r e s u l t i s remarkable. It does no t e x p l i c i t l y
c o n t a i n any jump c o n t r i b u t i o n s , as t h e s e are i m p l i c i t i n t h e r e p r e s e n t a t i o n
of t h e gene ra l i zed f u n c t i o n and its d e r i v a t i v e s i n terms of t h e o r d i n a r y
f u n c t i o n and i t s d e r i v a t i v e s , as given by t h e previous expres s ions (3-4)
t o (3-7).
It i s of g r e a t p r a c t i c a l i n t e r e s t t o determine t h e s impler expres s ions
t h e s e reduce t o i n t h e c a s e of a u n i l a t e r a l transform.
The t ransform of t h e d e r i v a t i v e of t h e o rd ina ry f u n c t i o n becomes
using (3-10)
where t h e Laplace i n t e g r a l i s taken w i t h a r igh thand lower l i m i t , and
t > 0. j
The u n i l a t e r a l t ransform of g ' ( t ) can be w r i t t e n as:
(3-14)
n
j =1 L ( g ' ( t ) ) = L { f ' < t > ) + c J { f ( t j ) ) Z s t j
+ f (O+) - f (0-)
fol lowing (3-7), w i t h t > 0. j
Eliminat ion of L ( f ' ( t ) ) between t h e two expres s ions (3-14), (3-15);
and i n s e r t i o n of L { f ( t ) } = L { g ( t ) ) and f(o-) = g(o-) l e a d s t o t h e
important r e s u l t :
L { g t < t > ) = s L { g w ) - g (0-1
(3-15)
(3-16)
It is i n t e r e s t i n g t o n o t e t h a t as was a l r e a d y pointed ou t be fo re ,
- 9-
t h e jumps do not e x p l i c i t l y appear i n t h i s formula, bu t are i m p l i c i t l y
contained i n g ( t > , and t h a t t h e i n i t i a l va lue appearing i n (3-16), is
t h e i n i t i a l va lue from t h e l e f t . Transform expressions f o r h ighe r o r d e r
d e r i v a t i v e s of o rd ina ry func t ions become p rogres s ive ly more unwieldy due
t o t h e presence of jumps i n t h e v a r i o u s d e r i v a t i v e s , b u t no such problem
e x i s t s w i t h general ized d e r i v a t i v e s . Successive s u b s t i t u t i o n s l e a d to :
I 2 L { g " ( t ) ) = s L { g ( t > ) - s 8 6 ) - g v o - ) (3-17)
and f i n a l l y f o r the d e r i v a t i v e of o r d e r n:
(n-l) (0-) (3-18) n L { g ( " ) ( t ) } = s L { g ( t ) } - sn-l g(0-1 - ..... -g
This expres s ion seems t o be n e i t h e r well-known, nor widely used, It i s
i n agreement, however, w i t h t h a t g iven by Zadeh and Desoer (11). I n
view of t h e importance of (3-18) i n a p p l i c a t i o n s , v a r i o u s a l t e r n a t i v e s
t o t h e preceding development l e a d i n g t o t h e same f i n a l r e s u l t w i l l be
explored i n t h e next s e c t i o n s .
I V METHOD BASED ON THE CONSTRUCTION OF THE EQUIVALENT CONTINUOUS FUNCTION
It i s p o s s i b l e t o c o n s t r u c t an equ iva len t cont inuous f u n c t i o n f c ( t )
from f ( t ) as represented i n F igu re 2 , by s l i d i n g t h e curve f 2 ( t ) , f , ( t > ,
and f ( t ) down by t h e jumps, J { f ( t l ) } , J { f ( t l ) } + J { f ( o ) ) and J { f ( t , ) }
+J ( f ( o ) ) + J { f ( t , ) } r e s p e c t i v e l y .
4
This g i v e s t h e e x p l i c i t expression:
Reca l l i ng t h a t f o r a continuous f u n c t i o n f ( t ) C
L2 {f',(t>) = s L { f c ( t > } 2
-10-
( 4 - 2 )
and t ak ing i n t o account t h e a c t i o n of t h e u n i t s t e p f u n c t i o n on t h e v a r i o u s
terms l e a d s t o
L { f r ( t ) } = s L [ f ( t ) - J { f ( t ) I u ( t - t l ) - J{f(o)} u ( t ) 2 2 1
( 4 - 3 ) -J {f (t )I ~ ( t - t ~ ) ]
2
It should b e noted t h a t no d i f f e r e n c e e x i s t s between f ' ( t ) and f:(t) when
they are considered as f u n c t i o n s i n t h e o r d i n a r y sense , s i n c e they are
l e f t undefined a t t h e p o i n t s where f ( t ) s u f f e r s a jump d i s c o n t i n u i t y .
The r e s u l t ( 4 - 3 ) can now be general ized t o :
which confirms t h e r e s u l t ( 3 - 1 0 ) prev ious ly obtained. I f t h e f u n c t i o n
is now considered as a gene ra l i zed f u n c t i o n i t can b e w r i t t e n as:
Upon d i f f e r e n t i a t i o n , g e n e r a l i z a t i o n and making u s e of f ' ( t ) = f i ( t ) , i t
fo l lows t h a t :
n
i-1 g ' ( t ) f ' ( t ) + C J { f ( t i ) } 6 ( t - t i )
Taking t ransforms of bo th s i d e s of ( 4 . 6 )
( 4 - 6 )
( 4 - 7 ) .
El imina t ion of f ' ( t ) between t h e two expres s ions ( 4 - 4 ) and ( 4 - 7 )
l e a d s , r e c a l l i n g t h a t L { g < t ) ) = L2 { f ( t ) } , t o t h e following: 2
-11-
This is p r e c i s e l y t h e expres s ion obtained i n (3-13), and from t h a t
p o i n t on t h e d e r i v a t i o n is as g iven i n Sec t ion 111.
where use has been made of a w e l l known theorem i n S t i e l t j e s i n t e g r a t i o n
i n t h e eva lua t ion of t h e second term (10). Combining (5 -2) wi th (4 -4)
y i e l d s
L 2 { g ' ( t ) ) = s L { g ( t > \ (5-3) 2
which i s t h e same as (4 -8 ) .
V I NETHOD BASED ON THE THEORY OF DISTRIBUTIONS
By d e f i n i t i o n t h e gene ra l i zed d e r i v a t i v e of a f u n c t i o n f ( t ) w i th
f i n i t e jumps can be de f ined as fo l lows i n t h e s e n s e d i s t r i b u t i o n s ( 4 )
-12-
- ~
~ ~~ ~
~~ ~ . -~
V METHOD BASED ON THE STIELTJES INTEGRAL
It is p o s s i b l e t o cons ide r t h e Bilateral Laplace Transform of t h e
d e r i v a t i v e of a f u n c t i o n e x h i b i t i n g f i n i t e jumps as t h e S t i e l t j e s i n t e g r a l ~
L 2 ( f ' ( t ) ) = ,Iw e'st d { f ( t ) ) (5 -1)
fol lowing Widder ( 1 2 ) .
I n s e r t i n g t h e expres s ion (4 -5 ) i n t o (5-1) r e s u l t s i n t h e following: n
o r r e w r i t i n g : I
< f , $1 > = - I - m f ( t ) $ ' ( t ) d t c f ' , $ > = -
where $(x) i s a t e s t i n g func t ion . P a r t i c u l a r i z i n g t h i s t o t h e f u n c t i o n
r ep resen ted i n F igu re 2, t h e i n t e g r a l can b e s p l i t over t h e a p p r o p r i a t e
domains, y i e l d i n g
I n t e g r a t i n g (6-2) by p a r t s i n o rde r t o t r a n s f e r a l l t h e d i f f e r e n t i a t i o n s
from $ ( t ) t o f ( t ) one ob ta ins :
+ 1:' f ' ( t ) $ ( t ) d t + I" f ' ( t ) $ ( t ) d t t 2
(6-3)
where u s e has been made of t h e f a c t t h a t t h e t e s t i n g f u n c t i o n van i shes
o u t s i d e a f i n i t e i n t e r v a l . Upon g e n e r a l i z i n g , i t fo l lows t h a t :
n
i= 1 SI, f ' ( t ) $(t) d t + C J { f ( t i ) } $ ( t i ) (6-4) c f ' , JI > =
o r denot ing by g ' ( t ) t h e general ized d e r i v a t i v e given by t h e expres s ion
(6-4)
n g ' ( t ) = f ' ( t ) + C J { f ( t i ) } 6 ( t - t i )
i=l
where use of t h e s i f t i n g property has bean made. Here t h e symbol f ' ( t )
-13-
s t a n d s f o r t h e d e r i v a t i v e of f ( t ) i n t e r p r e t e d i n t h e o rd ina ry sense,
wherever i t e x i s t s and i t can be a r b i t r a r i l y a s s igned any f i n i t e v a l u e where
i t does no t e x i s t .
Therefore the gene ra l i zed d e r i v a t i v e i s equa l t o t h e o rd ina ry d e r i v a t i v e
p l u s t h e jump c o n t r i b u t i o n s . It i s t h i s las t term t h a t i n s u r e s t h a t t h e
i n t e g r a l of g ' ( t > w i l l y i e l d t h e discont inuous f u n c t i o n f ( t ) r a t h e r t han
t h e equ iva len t continuous f u n c t i o n f c ( t ) t h a t would have been obtained
by cons ide r ing t h e i n t e g r a l of f ' ( t ) a lone. I n o t h e r words, t h e i n t e g r a t i o n
of g ' ( t ) au tomat i ca l ly in t roduces t h e c o r r e c t i n t e g r a t i o n cons t an t .
It should be noted t h a t equat ion (6-5) i s i d e n t i c a l t o (4-6) and i ts
t ransform is:
n L ( g ' ( t > ) = L2 { f ' ( t > ) + C J { f ( t i ) } s s t i (6-6)
i=l 2
On t h e o t h e r hand (4-4) g ives :
n
i=l (6-7) ' S t i
L { f ' < t > ) = s L ( f ( t ) ) - C J { f ( t i ) } e 2 2
El iminat ing f ' ( t ) between (6-6) and (6-7) and r e c a l l i n g t h a t L
L { g ( t > ) r e s u l t s i n
{ f ( t ) ) = 2
2
L { g ' < t > ) = s L { g w ) 2 2
The s p e c i a l i z a t i o n of t h i s r e s u l t t o t h e u n i l a t e r a l t r ans fo rm as w e l l as
i t s ex tens ion t o higher o rde r d e r i v a t i v e s have been developed i n Sec t ion 111.
. .
V I 1 SOLUTION AND DISCUSSION OF SOME ELEMENTARY PROBLEMS
1. Transform of t h e Space Var i ab le
Consider the problem of a simply supported beam sub jec t ed t o a con-
-14-
c e n t r a t e d load P as shown i n f i g u r e 3
'1 P
Figure 3
The Euler-Bernoul l i equa t ion f o r t h e d e f l e c t i o n of t h e beam i s
E 1 y ( 4 ) (x) = w(x)
where E 1 has been assumed constant . The u s u a l o p e r a t i o n a l s o l u t i o n
f o r t h i s beam is t o write w(x) = P 6(x-a)
EI y( '+) (x) = - P &(x-a>
T h i s equa t ion i s u s u a l l y transformed i n t o :
+ E1 [s4 y(s ) - s 2 y ' ( o ) - ylt ' (o+)] = - P zas
(7-2)
(7-3)
+ + where y (o ) = y"(o ) = 0 h a s been inco rpora t ed i n t o (7-3). y(x ) i s
now obtained by a s imple inve r s ion , and produces t h e expected r e s u l t .
The boundary c o n d i t i o n s y(L-) = y"(L-) = 0 are t aken i n t o account i n o rde r
t o determine y ' (o ) and y"' (0 ). Unfortunately, t h i s c o r r e c t r e s u l t i s due
t o t h e compounding of two e r r o r s t h a t cance l each o t h e r out .
y ( x ) is considered an o r d i n a r y funct ion, i t s t h i r d d e r i v a t i v e yl1'(x) w i l l
e x h i b i t a jump equa l t o t h e shear -P a t t h e p o i n t x = a, and i t s c o n t r i b u t i o n
must b e included i n t h e expression of t h e t ransform (3-14).
+ +
F i r s t , i f
This , however,
-15-
would l e a d t o the i n c o r r e c t s o l u t i o n i f t h e r ighthand s i d e of (7-2)
i s r e t a i n e d . It i s no t d i f f i c u l t t o see t h e r eason f o r which one would
no t o b t a i n t h e c o r r e c t s o l u t i o n . Consider t h e equa t ion
E1 y ' ' ' ( x ) = - Pb - P u(x-a> L (7 -4)
I f y (x ) i n t h i s equa t ion i s considered an o rd ina ry f u n c t i o n then t h e
d e r i v a t i v e i s E1 y ( 4 ) = 0.
and t h e r e f o r e not d i f f e r e n t i a b l e t h e r e , i t s va lue can be a r b i t r a r i l y taken as
ze ro ,
Since u(x-a) is discont inuous a t x = a ,
Taking t h e transform of y (x ) and i n s e r t i n g t h e a p p r o p r i a t e jump
c o n t r i b u t i o n y i e l d s :
Recall ing t h a t
+ and inco rpora t ing y (o ) = ~"(0 ' ) = 0 y i e l d s
This i s p r e c i s e l y t h e equa t ion obtained b e f o r e as a r e s u l t of neg lec t ing
t h e jump c o n t r i b u t i o n , and r e p l a c i n g i t by t h e d e r i v a t i v e of t h e u n i t
s t e p f u n c t i o n which should have been t aken as zero.
t h e n e t e f f e c t of t h e s e two e r r o r s a n n i h i l a t e each o t h e r , thereby
producing a c o r r e c t r e s u l t . Although the i n c l u s i o n of t h e jump due t o
t h e shear would s o l v e the p a r t i c u l a r problem under c o n s i d e r a t i o n , t h i s
procedure cannot admit any g e n e r a l i z a t i o n . The d e f l e c t i o n of t h e beam
It i s seen t h a t
-16-
can b e considered as t h e response of t h e beam t o t h e concen t r a t ed
load i n p u t P and t h e r e f o r e , should b e uniquely determined, once t h e
system parameters, i n i t i a l cond i t ions and t h e i n p u t are completely
known.
and should be r e f l e c t e d e x p l i c i t l y i n t h e expres s ion of t h e d e f l e c t i o n .
To i n c o r p o r a t e t h e jump cond i t ion i n t h e shea r i n t o t h e transformed
I n o t h e r words, t h e jump i n t h e shea r i s p a r t of t h e response
d e f l e c t i o n equat ion, is t o assume i n an a p r i o r i f a s h i o n t h e p a r t i a l
s o l u t i o n of t h e problem. It is not d i f f i c u l t t o see t h a t such
a n t i c i p a t o r y knowledge would b e much more d i f f i c u l t i n t h e case of
l i n e a r d i f f e r e n t i a l equat ions with v a r i a b l e c o e f f i c i e n t s , where t h e
l o c a t i o n and magnitude of t h e jumps i s n o t as easy t o a s c e r t a i n . The
theo ry of t h e Green's Funct ion would be of h e l p h e r e , b u t t h i s is
equ iva len t t o ob ta in ing t h e complete s o l u t i o n of t h e problem.
To t h e knowledge of t h i s writer, C h u r c h i l l (14) i s t h e only au tho r
t o have pointed ou t t h e i n c l u s i o n of t h e jump i n t h e shea r as a n
a l t e r n a t i v e t o t h e classical t reatment of t h e problem u s u a l l y given i n
books on Transform Theory. The l i t e r a t u r e seems t o g i v e no i n d i c a t i o n as
t o t h e e f f e c t t h a t t h e c l a s s i c a l t r ea tmen t is i n c o r r e c t o r t h a t t h e in-
t r o d u c t i o n of t h e concept of a gene ra l i zed f u n c t i o n i s necessary t o t rea t
t h e problem s a t i s f a c t o r i l y .
+ Rewriting t h e equa t ion f o r t h e shea r i n 0- < x < L as:
Pb u(x) - P u(x-a> + - P a u (x-L) L E I Y"'(x) = - L
I f y(x) is now considered a gene ra l i zed func t ion , i t s d e r i v a t i v e
becomes
6 (x-L) (4) Pb P a E 1 y (XI = 6(x) - P 6(x-a) + - L
(7-7)
(7-8)
-17-
Clea r ly , s i n c e r ighthand s i d e i s a gene ra l i zed f u n c t i o n , t h e l e f t h a n d
s i d e has t o b e considered as a gene ra l i zed f u n c t i o n and t r e a t e d as such
i n i t s transform. For t h e sake of c l a r i t y , i f t h i s gene ra l i zed f u n c t i o n
i s denoted by y (x) , one
accordance w i t h (3-18).
g
EI [s4Yg(s) - s3yg(o-)
o b t a i n s t h e fo l lowing transformed equa t ion i n
Pb Pa ;Ls L
= - - peas + ( 7 - 9 )
It should be noted t h a t t h e r e a c t i o n a t t h e l e f t w a s introduced,
s i n c e lef thanded l i m i t s now make t h i s r e a c t i o n p a r t of t h e inpu t .
C lea r ly , i f one cons ide r s t h e beam as extending beyond t h e l e f t support
y" (0-) = y' " (0-) s i n c e t h e bending moment and shea r vanish along t h e
ex tens ion of t h e beam t o t h e l e f t , which is a s t r a i g h t l i n e . I n a d d i t i o n ,
i f t h e o r i g i n is taken t o co inc ide wi th t h e l e f t support y(o-) = 0.
Therefore t h e equat ion reduces to:
Pa -Ls +- e - - y ' ( o - > + - - - - 1 P b P zas S ~ E I AIL Y,(S) - EIL
S2 S4
I n v e r t i n g
Pa<x-p 6EIL
P b x 3 - P < x - ~ > ~ + 6EIL 6EI
where < ~ - a > ~ i s the Macaulay Bracket n o t a t i o n f o r (x-a)3 u(x-a), and
t h e l a s t term has no c o n t r i b u t i o n t o t h e d e f l e c t i o n u n l e s s t h e beam
extends beyond the r i g h t support .
The boundary c o n d i t i o n y(L) = 0 s e r v e s t o determine t h e only
(7-10)
(7-11)
unknown y'(o-) remaining i n t h e equation.
is:
I n s e r t i n g t h i s v a l u e which
Pb (b2-L2) 6EIL y'(o-) = -
i n t o t h e d e f l e c t i o n equa t ion y i e l d s :
L Y W = b
Pb [ (b2-L2) x + x 3 - - < ~ - a > ~ ]
which produces t h e jump i n t h e shear as p a r t of t h e response, w i thou t
any such a p r i o r i assumption.
2 . Transform of t h e T i m e Variable
Consider t h e equa t ion of motion of a harrtionic o s c i l l a t o r
X ( t ) + p2x( t ) = 0
(7-12)
(7-13)
(7-14)
where p2 = k/m s u b j e c t t o i n i t i a l c o n d i t i o n s x(o-) = &(o') = 0 and
s t r u c k by a blow of impulse Io. The problem can b e handled i n two
d i f f e r e n t ways via t h e Laplace Transform.
L e t x ( t ) be an o r d i n a r y funct ion. Then
% ( t ) + p 2 x ( t ) = 0 (7-15)
I
m + * + Sub jec t t o x(o ) = 0 x ( 0 ) = - where t h e second c o n d i t i o n has been
ob ta ined by a c o n s i d e r a t i o n of l i n e a r
is
I m 0 1 E(s) =
s2 + p2
momentum. The transformed equa t ion
a f t e r i n s e r t i o n of i n i t i a l cond i t ions , i t fol lows t h a t
I - 0 x ( t ) = -
m P s i n p t t > O
(7-16)
(7-17)
-19-
L e t x ( t ) now be a gene ra l i zed f u n c t i o n . The equa t ion of motion
can now b e w r i t t e n as:
s u b j e c t t o x(o-) = i ( o ' ) = 0.
The transformed equa t ion now r e a d s
I o / m
s 2 + p2 f ( s ) =
and i t s i n v e r s e i s 1 A 0 x ( t ) = -
m P s i n p t u ( t )
It i s easy t o show t h a t t h i s s o l u t i o n s a t i s f i e s t h e g iven equa t ion
and t h e i n i t i a l c o n d i t i o n s e i t h e r by formal s u b s t i t u t i o n o r by a s imple
a p p l i c a t i o n of d i s t r i b u t i o n theory.
A number of textbooks such as (13) use t h e equa t ion
(7-18)
K ( t ) + p 2 x ( t ) = - m
(7-19)
(7-20)
(7-21)
+ * + w i t h r i g h t s i d e d i n i t i a l c o n d i t i o n s x(o ) = x ( o ) = 0, and e x p l a i n t h e
discrepancy between t h e assumed zero i n i t i a l v e l o c i t y and t h e a c t u a l
i n i t i a l v e l o c i t y I /m by appeal ing t o momentum c o n s i d e r a t i o n s .
t h a t no c o n t r a d i c t i o n a c t u a l l y e x i s t s , and t h e apparent c o n t r a d i c t i o n so
It is c l e a r 0
f r e q u e n t l y encountered i n t h e l i t e r a t u r e can be e a s i l y r e so lved by cons ide r ing
gene ra l i zed f u n c t i o n s which a u t o m a t i c a l l y e x h i b i t l e f thanded l i m i t s i n
t h e transforms of t h e i r d e r i v a t i v e s .
-20-
~
VI11 CONCLUSIONS
It has been shown that the failure to recognize the difference between
an ordinary function and a generalized function, particularly of their
derivatives, and transforms of derivatives; has led to well-known discrepancies
between the assigned initial values and the actual initial values.
has been discussed at length in the literature by several authors such
as (16).
generalized functions or distributions developed by Laurent Schwartz ( 4 ) .
This
This difficulty can be avoided by the use of the concept of
An elementary derivation of the transform of the generalized derivative
is developed in this paper and applied to typical problems in which it is
shown that some of the classical contradictions disappear. It is hoped
that a more widespread understanding of these relatively new concepts by those
interested in applications, will give them a more powerful method of approach
to boundary value problems, particularly those leading to partial differential
equations.
REFERENCES
1. Wylie, C.R., Jr., 11 Advanced Engineering Mathematics," Third Edition,
McGraw-Hill, 1966, p. 236, problem 3.
2. Thomson, W.T., "Laplace Transformation," Second Edition, Prentice-Hall,
1960, p. 7, problem 16.
3. Van der P o l , B. and Bremmer, H., "Operational Calculus Based on the
Two-sided Laplace Integral," Cambridge University Press, 1955.
4. Schwartz, L., "Mathematics for the Physical Sciences," Addison-Wesley,
1966, p. 80.
5. Boley, B.A., "Discontinuities in Integral Transform Solutions," Quarterly
of Applied Mathematics, Vol. 19, No. 4, January 1962.
6 . Reid, W.P., "On Some Contradictions in Boundary Value Problems,"
Mathematics Magazine, Vol. 37, No. 3, May 1964.
7. LePage, W.R., "Complex Variables and the Laplace Transform for
Engineers ,I1 McGraw-Hill, 1961.
8 . Hobson, E.W., "The Theory of Functions of a Real Variable," Volume I,
Cambridge University Press, 1927, p. 308.
9. Rasof, B., "The Laplace Transform of the Sectionally Continuous
Derivative of a Sectionally Continuous Function," Journal of the
Franklin Institute, Vol. 277, No. 2, February 1964, pp. 119-127.
10. Stakgold, I. , "Boundary Value Problems of Mathematical PhYsics," Vol. I, Macmillan, 1967, p. 39.
-22-
I I'
11. Zadeh, L.A., and Desoer, C.A., "Linear System Theory, The State
Space Approach," McGraw-Hill, 1963, p. 540.
12, Widder, D.V., "The Laplace Transform," P r ince ton Un ive r s i ty P res s , 1941.
13, Danese, A.E., "An In t roduc t ion t o Applied Mathematics," Al lyn and Bacon,
1965, p. 721.
14. Churchi l l , R.V., "Operat ional Mathematics," Second Ed i t ion , M c G r a w - H i l l ,
1958, p. 94, problem 5.
15. Hildebrand, F.B., "Advanced Calculus f o r Appl ica t ions ," Prent ice-Hal l ,
1963, pp. 68-69.
16. Doetsch, G., "Guide t o t h e Appl ica t ions of Laplace Transforms,"
D. Van Nostrand, 1961, pp. 94-105.
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Lnclassified . - - _ ~
Securiti Cldssificatlon
1.1 KEY UORDS. Ke? u o r d s a r e !echnica l l ) r e a n i n g f u l :e rn o r (ha! c b , ~ r a c : e r i z e a repor: and -a). be used a index f e r c a t a l o j i n ~ :he repsr! Key H s r d s -US: b e select6.2 F,: rha: 3.- =c:.i:L:\ c la s s : f : ca : :on 15 :eq,.:i:ed, Iden. e l e r s , I ; uc , as e q L l p - e n : -:de: 6 e s i g n a t i o n . t rade 7.a-e - 1 1 1
I I I N S T W C T I O S S
1. ORIGINATING ACTIVITY: Enter t h e name and a d d r e s s of t h e cont rac tor , subcontractor , g ran tee , Department of De- f e n s e ec t iv i t y or o ther organiza t ion ‘corporate eufhor) i s s u i n g t h e repofl . 2a. R E P O R T S E C U A T Y CLASSIFICATION: E n t e r t h e over- a l l s e c u r i t y c l a s s i f i c a t i o n of the report. I n d i c a t e whether “ R e i t r i c t e d Data“ is included. a n c e with appropr ia te secur i ty regulat ions. 2b. GROUP: Automatic downgrading i s s p e c i f i e d i n DoD Di- r e c t i v e 5200.10 a n d Armed F o r c e s Indus t r ia l Manual. Enter the group number. Also . when a p p l i c a b l e , show tha t op t iona l markings h a v e b e e n used for Group 3 and Group 4 a s author- ized.
Marking is to b e i n accord-
L I N K C
R O L E .‘*-
10. AVAILABILITY LIMITATION NOTICES E n t e r a n ) lim- i t a t i o n s on further d i s s e m i n a t i o n of the report o ther than thos imposed by securi t ! c l a s s i f i c a t i o n , us ing s tandard s ta tements s u c h a s
(1) “Qual i f ied r e q u e s t e r s ma) obtain c o p i e s of t h i s report from D D C ”
(2) “ F o r e i g n announcement a n d d isseminat ion of t h i s report b ) DDC is not authorized. ”
(3) “C S G o \ e r n F e n t a g e n c i e s nab obta in c o p i e s of t h - s report d u e c t l ) from DDC. u s i r s s h a l l reqLest through
Other qual i f ied DDC
t t - 3, R E P O R T T I T L E Enter the comple te report t i t l e i n a l l (4) “L. S rrJlitar) a g e n c i e s may obta in c o p i e s of t h i s c a p i t a l l e t te rs . T i t l e s in a l l c a s e s should b e unc lass i f ied . If a meaningful t i t l e cannot be s e l e c t e d without c l a s s i f i c a - t ion , show t i t l e c l a s s i f i c a t i o n in a l l c a p i t a l s i n p a r e n t h e s i s immediately fol lowing the t i t le .
4. D E S C R I P T I V E NOTES If appropriate , e n t e r the type o f , (5) “ 1 1 1 distribution of t h i s report is controlled. @d- report , e. g., in ter im, progress , summary, annual , or f inal . G i v e the i n c l u s i v e d a t e s when a s p e c i f i c repor t ing per iod IS 1
5 . AUTHOR(S) or i n t h e report . If mil i tary, show rank and branch of serv ice . T h e name of t h e pr inc ipa l author is an a b s o l u t e minimum r e q u r e m e n t .
report direct ly from D D C st-all r e q u e s t through
Other qual i f ied u s e r s
i f ied DDC u s e r s sha l l reques t through # O
I f t h e report h a s b e e n furrushed to t h e Off ice of Technica l Enter t h e name(s) of au thol i s ) a s shown o n 1 services, ~ ~ ~ ~ ~ t ~ e n t o f Commerce, for s a l e to t h e publ ic , ind Enter l a s t name, f v s t name, middle ini t ia l . , c a t e this fac t and e n t e r t h e p r i c e , i f known. , 11. S U P P L E M E N T A R Y KOTES. U s e for addi t iona l explana-
covered. 1
~ tory notes . , 12. SPONSORING MILITARY ACTIVITY
~ ing for) t h e r e s e a r c h and development-
6. R E P O R T DATE. Enter t h e d a t e of the report a s day , month, year, or month, year, on t h e report , u s e d a t e of publ icat ion. 7a. T O T A L NUMBER OF PAGES, T h e to ta l p a g e count should fo l low normal paginat ion procedures , i.e., enter t h e number of p a g e s contaming information
7 b . NUMBER OF R E F E R E N C E S Enter t h e to ta l number of If addi t loma\ SDace i~ required a cont inua t ion shee : re ferences c i t e d m t h e report. 8 a C O N T R A C T OR GRANT NL’MBER If appropr ia te , e n t e r the a p p l i c a b l e number of t h e cont rac t or grant under which t h e report w a s written.
If more than o n e d a t e appears Enter the name of t h e depar tmenta l p ro jec t o f f ice or labora tor ) sponsor ing ( p a y
13 ABSTRACT Enter an a b s t r a c t g ~ \ l n % a br ief and f a c t u a l s u n m a r ) o r the d o c u v e n t i n d l c a t i \ e of the report i t r ra \ a l s o a p p e a r e l s e \ \ h e r e In the bod\ O f the technica l re-
Inc lude a d d r e s s ,
exen thougt
s h a l l b e a - [ a c h e d .
ports b e u : i c i a s s i f i e d . Each p a r a v a p h of :he abs t rac t s h a l l end \vi:h a n in . l lca! icn of :he - i l l tar) Sec’uri!) C~asslf lCat iOn o f t h e :nfc:-a:ion :n t h e paraizraph represen!ed a s ‘TS! ‘s;
It IS h i g h l y des i rab :e tha: :he abs t rac! of c l a s s i f i e d re-
’C). or ‘L’
e \ e r tile . ; u g g e s . e ~ \enp:-. :s i r 3 - 15 ’1 :t 2 2 5 ivcrds. T h e r e . s nc !i-i:ati:3 - n !he IenGh of :‘.e absrrac: How,
1 I apt iona l
a -
# .
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