From "Fractional Calculus and its Applications", Springer Lecture Notes in Mathematics,
volume 57, 1975, pp.1-36.
A BRIEF HISTORY AND EXPOSITION OF THE FUNDAMENTAL THEORY
OF FRACTIONAL CALCULUS
BERTRAM ROSS
Abstract:
This opening lecture
is intended to serve as a propaedeutic
for the papers to be presented at this conference whose nonhomogeneous
audience
includes scientists, mathematicians,
This expository and developmental
lecture,
engineers
and educators.
a case study of mathemati-
cal growth, surveys the origin and development of a mathematical
from its birth in intellectual
curiosity to applications.
mental structure of fractional
calculus is outlined.
for the use of fractional
cated.
calculus
idea
The funda-
The possibilities
in applicab]e mathematics
is indi-
The lecture closes with a statement of the purpose of the con-
ference.
Fractional
calculus has its origin in the question of the ex-
tension of meaning.
of real numbers
A well known example is the extension of meaning
to complex numbers,
meaning of factorials of integers
and another is the extension of
to factorials of complex numbers.
In generalized integration and differentiation the question
extension of meaning is:
order
dny/dx n
irrational,
be extended to have meaning where
fractional
Leibnitz
invented the above notation.
that prompted L'Hospital
possibility that
n
be a fraction.
Leibnitz
Perhaps,
n
infinite product for
it was naive
be ½?", asked
"It will lead to a paradox."
"From this apparent paradox,
ful consequences will be drawn."
that differential
is any number---
to ask Leibnitz about the
"What if
[i] in 1695 replied,
But he added prophetically,
Wallis's
n
or complex?
play with symbols
L'Hospital.
of the
Can the meaning of derivatives of integral
one day use-
In 1697, Leibnitz, referring to
~/2, used the notation
d2y
and stated
calculus might have been used to achieve the same
result.
In 1819 the first mention of a derivative of arbitrary order
appears
in a text.
The French mathematician,
S. F. Lacroix
[2],
published
a 700 page text on differential
which he devoted less
than two pages
Starting with
n
a positive
symbol
and by replacing
m
he found the
F
by
in the manner typical
Lacroix obtained
mth
and
n
x a.
formalists
factorial,
real number
a,
of this period,
the derivative
of arbitrary
for
(x)
=
dx ½
= ½F(½)
yielded by the present
xa-½
r (a+½)
He gives the example
F(3/2)
the generalized
by any positive
of the classical
to be
the formula
d½
because
derivative
n -m
which denotes
1/2
dx ½
tion
in
x
d2y = F(a+l)
which expresses
calculus
y = xn~
integer,
dmy _
n!
dx m
(n-m) !
Using Legendre's
and integral
to this topic.
order
y = x
1/2
of the func-
and gets
2~
/-~
= ½/-# and
F(2) = i.
day Riemann-Liouville
279 years
This
result
definition
tional
derivative.
It has taken
raised
the question
for a text to appear solely
is the same
of a frac-
since L'Hospital
devoted
first
to this topic~
[3].
Euler and Fourier made mention
order but
making
they gave no applications
the first application
Abel applied
the fractional
equation which
This problem,
arises
belongs
calculus
wire
of the tautochrone
is placed.
of the wire
in the s a m e
The brachistochrone
time
problem
regardless
problem.
is that of find-
lying in a vertical
plane
that the time of slide of a bead placed on the wire slides
lowest point
of
[4] in 1823.
in the solution of an integral
called the isochrone problem,
ing the s~hape of a frictionless
of arbitrary
So the honor
to Niels Henrik Abel
in the formulation
sometimes
of derivatives
or examples.
such
to the
of where
the bead
deals with the shortest
time
of slide.
Abel's
attracted
attempt
solution was so elegant
the attention
of Liouville
to give a logical
definition
that it is my guess
it
[S] who made the first major
of a fractional
derivative.
He
published
three long memoirs
Liouville's
tives of integral
in 1832 and several more through
starting point
is the known result
1855.
for deriva-
order
Dme ax = ame ax
which he extended
in a natural way to derivatives
of arbitrary
order
DYe ax = aVe ax
He expanded
the function
f(x)
in the series
9o
(1)
f(x)
:
I
cn e
anX,
n=O
and assumed
the derivative
of arbitrary
order
f(x)
to be
co
(2)
DVf(x)
Cn a v e anx
=
n=O
This formula
obvious
is known
disadvantage
the series
as Liouville's
that
v
[6] first definition
must be r e s t r i c t e d
and has the
to values
such that
converges.
Liouville's
of the form
second m e t h o d was applied
x "a, a > O.
(3)
I
The transformation
He considered
=
f
xu = t
(4)
x-a _
to explicit
functions
the integral
ua-le-XUdu.
gives the result
1
I.
r(a)
Then, with the use of (I) he obtained,
of (4) with
after operating
on both
sides
D v, the result
(5)
DVx -a = (-l)Vr(a+v)
x -a-v
[7]
r(a)
Liouville
problems
was successful
in potential
theory.
in applying these definitions
"These concepts
last," said Emil Post
[8].
certain values
and the second method
class
of
of functions.
v
were too narrow
The first definition
is restricted
is not suitable
to
to
to
to a wide
Between 1835 and 1850 there was a c o n t r o v e r s y w h i c h c e n t e r e d
on two definitions
favored Lacroix's
mathematicians
[I0]
of a fractional
derivative.
George Peacock
g e n e r a l i z a t i o n of a case of integral order.
favored Liouville's
definition.
Augustus
[9]
Other
De M o r g a n ' s
judgement p r o v e d to be accurate when he stated that the two
versions
may very p o s s i b l y be parts of a more general
W i l l i a m Center
versions
of a fractional
tive of a constant.
fractional
while
In 1850
derivative
focused on the fractional
deriva-
A c c o r d i n g to the P e a c o c k - L a c r o i x v e r s i o n the
derivative
of a constant yields
according to Liouville's
of a constant equals
formula
zero because
The state of affairs
cleared up.
system.
[ii] observed that the d i s c r e p a n c y b e t w e e n the two
a result other than zero
(5) the fractional derivative
r(o) = ~.
in the m i d - n i n e t e e n t h
Harold Thayer Davis
[12] states,
century is now
"The m a t h e m a t i c i a n s
at
that time were aiming for a p l a u s i b l e definition of g e n e r a l i z e d differentiation but,
in fairness
to them, one should note they lacked
the tools to examine the consequences
of their definition
in the com-
plex plane."
Riemann
posthumously
[13]
in 1847 while a student wrote a paper p u b l i s h e d
in which he gives
a definition of a fractional operation.
It is my guess that Riemann was i n f l u e n c e d by one of Liouville's
memoirs
in which
Liouville wrote,
"The ordinary differential equation
dny = O
dx n
has the complementary
solution
Yc = Co + ClX + c 2 x 2
+ "'" + Cn-I x n - 1
Thus
du
,
f(x)
=
o
dx u
should
have
clined
to
a corresponding
believe
Riemann
complementary
saw fit
his definition of a fractional
(6)
Cayley
D-v
f(x)
=
1
r (v)
to
add
solution."
So,
a complementary
I am i n function
integration:
(x-t)v-lf(t)dt
+ ,(x).
;c
[13] remarked in 1880 that Riemann's
is of indeterminate nature.
complementary
function
to
The development
Peacock made several
he m i s a p p l i e d
which
of mathematical
errors
in the topic of fractional
the Principle of the Permanence
is stated for algebra and which
theory of operators.
in his discussion
Liouville made
of one of the parameters
versions
sults when
Heaviside
century,
Riemann became hope-
Thus,
I suggest
silence
Two
different
re-
that when Oliver
of the nineteenth
and disdain not only because
jibes he made at mathematicians
the distrust mathematicians
function.
yielded
in the last decade
he was met with haughty
of the hilarious
apply to the
complementary
derivative
applied to a constant.
when
function that the specialization
led to an absurdity.
p u b l i s h e d his work
calculus
of Equivalent Forms
did not always
an indeterminate
of a fractional
error.
an error when he failed to note
of a complementary
lessly entangled with
different
ideas is not without
had in the general
but also because
of
concept of fractional
operators.
The subject
ness of notation
papers
that
notation
of notation
of fractional
follow
calculus
in this text,
I prefer was
cannot be minimized.
The succinct-
adds to its elegance.
various
notations
invented by Harold T. Davis.
In the
are used.
The
All the informa-
tion can be conveyed by the symbols
cD~ v f(x),
denoting
integration
scripts
c
and
a definite
x
of arbitrary
denote
integral
to avoid ambiguities
becomes
integration
itly formulate.
fractional
a vital part
the mathematical
fractional
or complex,
wide
formalizing
f(z)
z, the derivative
frac-
but were try-
class,
for every function
and every number
D v f(z)
= g(z)
c z
If
symbol
problem of defining
a function
'
i.
The adjoin-
of the operator
Briefly what is w a n t e d is this:
D v f(x) = g(x)
when
z is purely
cx
to the following criteria:
able
integration.
of
they well understood but did not explic-
z = x + {y, of a sufficiently
irrational
The sub-
of integration
It is clear that the mathe-
so far were not merely
ing to solve a p r o b l e m which
f(z),
along the x-axis.
(terminals)
and differentiation.
mentioned
O,
in applications.
We now consider
tional
order
the limits
which defines
ing of these subscripts
maticians
v
real
is an analytic
cDzv f(z)
v,
or
'
should be assigned subject
function of the complex vari-
is an analytic
function
of v and z.
2.
as o r d i n a r y
is
The operation
a negative
same
result
along
with
D v f(x)
integer,
as
its
3.
n-1
when
say
ordinary
The operation
at
C
D- n
X
the same result
integer.
f(x)
and
C
D- n
X
If
v
must
produce
f(x)
must
the
vanish
x = c.
zero leaves
the function
un-
= f(x)
X
The fractional
operators
cDx v [af(x) + bg(x)]
must be linear:
= a cD-Vx f(x)
The law of exponents
5.
a positive
then
of order
DO f ( x )
4.
is
integration
derivatives
C
v
v = -n,
n-fold
changed:
must produce
C X
differentiation
+ b cD-Vx g(x)
for integration
of arbitrary
order holds:
cDxu- cD-Vx f(x)
A definition
of Riemann
(7)
This definition
we have
Riemann's
to establish
definition
is discussed
The definition
in honor
(x-t)v-lf(t)dt"
of arbitrary
and when
(see
the above
named
fX
order
stated
is the same as
function.
c = -= , (7)
[6], pp. 176-178).
a set of criteria
This question
criteria
but has no complementary
definitions
to fulfill
these
= irtvjlr
~ jc
for integration
definition
Liouville's
criteria,
that will
When
c = O
is equivalent
Although
it might be of interest
characterize
later in this text p.
(7) can be obtained
(7) uniquely.
379.
in at least
~0 x (x-t)bt d dt = F(b+l) F(d+1)_ xb+d+l
F(b+d+2)
(s)
b = 3
four differ-
and
d = 4, (8) gives
r
b and d > -I
the result
(4)
x s
8.7-6 -5
If one were
constant
to
(7) can be
Euler had shown that
ent ways.
For
fulfills
is
cD-Vx f(x)
Riemann's
shown
which
and Liouville
= cD-U-Vx f(x)
to integrate
of integration
the function
x4
four times
each time to be zero,
and take the
the result will be
x8
1
8-7"6"5
Inquisitive
experimentation
of this type might
lead one to guess that
the above two results may be connected by the expression:
x4 =
-
oDx 4
fox
i
F (4)
(x-t) 3 t 4 dt,
or in general
(9)
oD; n f(x)
~OX
1
=
?(n)
The above is generalized by letting
The same r e s u l t
integral
iterated
( x - t ) n-1 f ( t )
n = v.
can be o b t a i n e d by c o n s i d e r i n g
- 2
F(x) = ~cx dx 1 ~cxl d x 2 - ' This
iterated
integral
~c xn
region
as a single
f(Xn) dXn '
integral by the
that is, by integrating
[14].
the n - f o l d
Xn - 1
f
dXn-1 Jc
can be written
method devised by Dirichlet,
priate triangular
dt.
The result
over an appro-
is
rX
F(x) =
If we denote
gration
1
~
r(n)
Jc
(x x
- n)
the operators
as
We may write
F(x)
ing by replacing
and
= cD; n f(x).
n
with
v
A third approach
linear differential
Then letting
xn = t
we again arrive
at (7).
dn - 1
dxn-I + ... + Pn(X)
+ Pl(X)
operator whose
are continuous
on some closed finite
Po(X)
I.
H
and generaliz-
to (7) may be deduced using the theory of
dn
Let
and of inte-
dx,
Let
be a linear differential
on
~c x
. . . .
equations.
L = Po(X) dx n..
> O
f(Xn) dxn "
of differentiation
D; 1
Dx
n-I
coefficients
interval
be the one-sided
Pk' 0 ~ k ~ n
I = [a,b]
Green's
and
function
for
L.
Then if
in
f
is any function
I, then for all
continuous
on
I, and
xo
is any point
x C I,
g(x)
=
f
H(x,~) f(~)d~
o
is
the
fies
solution
of the
the boundary
nonhomogeneous
equation
Ly = f ( x )
which
g ( k ) ( x o) = 0
[For further details
see,
,
0 ~ k ~ n-1
for example,
.
K. S. Miller,
Linear Differ-
ential Equations in the ReaZ Domain, W. W. Norton and Co.,
York
(19631;
satis-
conditions
Chapter
Inc., New
3.]
The Green's
function
H
is given explicitly
by
el(X)
qb2 (x)
...
~bn(X)
qbl (~)
¢2(~)
..-
(~n(~)
(-i) n-I
H(x,{) =
where
and
Po(~)W(~)
{¢k[l ~ k # n}
W
is a fundamental
set of solutions
of
Ly = O,
is their Wronskian.
Now s u p p o s e
L = Dn
Then
{l
Dny = O
,
x, x 2 ,
..., x n-l)
dn
dx n
is a fundamental
set of solutions
and
¢i(~)
w(~)
¢2(¢)
--.
:
i
(~)
"'"
Cn(~)
of
=
1
¢
¢2
0
1
2¢
O
O
2
o
o
o
(n-l)
. .
cn -i
(n-l) ¢n-2
(n-l) (n-Z) ¢ n-3
( n - I) !
!!
where
n-I
(n-l) !! = II k!
k=O
Thus
in this
special
case
2
1
X
X
•
1
¢
¢2
. .
0
1
2¢
o
o
o
•
•
X
n-i
cn-i
(-i) n-I
H(X,¢)
is a p o l y n o m i a l
=
( n - l ) !!
of degree
n-i
in
(-1) n-1
_ _
[(_l)n+l(n_2)
( n - l ) !!
x
leading
1
(n-l) !
But
H(x,¢)
3xk
Thus
¢
= O , 0 < k < n-2
•
x=¢
is a zero of m u l t i p l i c i t y
H(x,¢)
(I0)
Hence
if
xo
n-I
1
-I
- (n-1) l(x-¢)n
=
a,
(n-l) ¢n-2
(n-l) ! ¢
with
! !] =
•
and
coefficient
10
rx
(11)
1
g(x)
is the unique
Ja (x-~)n-I
~ (n-l')!
solution of the differential
f(~)d~
equation
dny - f(x)
dx n
which
assumes
write
(II) as
the initial
values
We may
~aX
aDxnf(x)
Now,
0 < k < n-l.
g~k~(a)r
~
= O,
=
of course, we replace
1
r (n)
n
(x_~)n-lf(~)d~
by
v
(with
.
Re ~ > O)
in the above
formula~ [15].
The fourth method of arriving
tour integration
these generalized
tegral
operators
in standard works
in securing
in 1888 used a contour
in 1890.
Laurent
Cauchy's
integral
formula
n
But here
To keep
three methods
values
i/(t-z) v+l
the function
semi-infinite
is
n!
U
no longer contains
starts
because
c,
cut be the
to negative
Let
C
infinity on
be the open contour
c < x, on the lower edge of
the real axis to A, around the circle
in the positive
the upper edge of the cut.
of
v! = F(v+l).
a pole but a branch point.
we let the branch
t = x > O
a the point
the cut, then goes along
(7) the generalization
no difficulties
single valued,
at
f(t)
dt.
(t-z) n+l
= 2--~-~-~J
ic
the real t axis as in the figure below.
It-x I < ~
at the origin as did A. Krug
of obtaining
creates
line starting
(or loop) which
P.A.
loop.
f(n) (z) = D z f(z)
to fractional
starting
theory.
in-
only passing
in 1884 used a contour that started and ended at -~,
now called a Laurent
In the previous
fact that
for themselves
in complex variable
Nekrassov
n
(7) is by con-
It is a curious
and their connection with the Cauchy
formula have succeeded
references
at definition
in the complex plane.
sense to B, and then back
to
c
along
ii
Im(t)
t-plsne
cut
of
n
Generalizing
gives
B/~ex
~.C
,Re(t)
the Cauchy integral formula to arbitrary values
F(v+I)
D v f(z) = " 2 ~
(t-z)
-v-lf(
t) dt
where we define
(t_x) -v-I = e(-V-l)in(t-x),
and where
In(t-x)
is real when
t-x
is a positive
By standard methods of contour
to (7) (see
integration we are a~ain led
[6] pp. 198-202).
The general validity of definition
positive
real number.
(7) for
integer,can be established by mathematical
we are concerned with criterion
2 which stipulates
must produce the same result as ordinary
v = n, n
induction.
a
Here
that the definition
integration.
There is no
loss of generality by taking the lower limit of integration to be
zero.
We have
1
ODxnf
~ fx) = F(n)
The above is obviously
oD if(x)~
(x_t)n-lf(t) dt.
true for
=
fo
n = i, for
f(t) dt.
Now assume the formula true for
n = k:
12
1
oOx- k f<x) o r(k)
Replace
k
with
~0X( x _ t ) k - l f ( t )
k+l :
oD; (k+l) f(x)
=
SO X ( x _ t ] k f l t ]
1 ...........
kr (k)
Operate on both sides of the above with
(12)
ODxk-
Applying
Leibnitz's
f(x)
d
= 77
1
(x
= O, and
g(x,x)
= 0
of an integral
~ (x_t)kf(t)
is the integrand
The last two terms
d(O)/dx
(x-t)kf(t)dt.
fox
1
- kr(k)
d(O)
g
and we have
~}(k)flo
g(O,x) - 7 T the function
dt.
oDx = d/dx
rule for the derivative
0Dx- k f ( x )
where
dt.
+ g(x,x)
(x-t)kf(t)
gives
dt
dx
--~
,
in (12).
on the right above vanish because
because
of
the
factor
(x-x).
T h e n we
rx
have
oDx-k f(x) =
k = n
and since
k
Io (x-t)k-lf(t)dt'
kr('~'i
we have by mathematical
induction
fX
-
1
oDxn f ( ~ ) = r(~"i
This
result
/
o
(x_t)n-lf(t)dt.
is the same as (9) obtained heuristically.
The definition
for differentiation
of arbitrary
be shown later to be an integration
followed by ordinary
tion.
criterion
It follows
that
(7) fulfills
order will
differentia-
2 for differentiation
and integration.
Criterion
3 states
the function unchanged,
whether
D v f ÷ f as
C
the
x
The investigation
that is,
cDx0 f(x) = O.
v ÷ 0 shows a concern with the continuity
x
D -v operator
C
that the operation of order zero leaves
at
v = O.
We have
of
of
13
-X
(13)
c DO
x f(x)
as a consequence
be taken
general,
O-~ .
- F(O)
1
of letting
~£ ( x - t ) O-1 f ( t ) d t ,
v = O
in
(7).
The f a c t o r
1/r(O)
can
equal to zero because
F(O) = ~ .
The i n t e g r a l
would, in
be divergent
a n d we h a v e t o d e a l w i t h t h e i n d e t e r m i n a t e
form
There are
several
We assume
ways of handling
f(t)
this
is e x p a n s i b l e
situation.
in a T a y l o r ' s
series w i t h
remainder:
f(t)
and t a k i n g
= f(x)
limits
lim
v÷O
+ (t-x)f'(x)
of b o t h
sides
oDxVf(x)
=
+ (t-x)2f"(x)/2i
of (7) as
[Jo
lim
v÷O
~Ox ( x - t ~ , , v f ' ( x ) d t + - - "
+ (_l)n+ 1 f(n+l)(@)
(n+l) I
except
~OX
l i r a r ( v ) = ~ , a l l t h e t e r m s on t h e
v+O
the first
because the first
integral
,
we have
(x-t)v-1
F (v)
f(x) dt
;ox ( x n_ t!)r (vv÷)n - 1
+ (-1) n
r(v)
Because
v÷O
+ ---
f(n) (x)dt
(x_t)v+n
F (v) ....... d t
right
has
above vanish
the value
xVf(x)
r (v+l)
due to the g a m m a
no
function
loss of g e n e r a l i t y
of
relation
u s i n g the
vF(v)
= r(v+l).
We n o t e
l o w e r limit of i n t e g r a t i o n
there
O
instead
c.
Thus
lira
v÷O
oDx v f(x)
D O f(x)
Ox
We
can also arrive
= lira xVf(x)
v÷O
F (v+l)
= f(x)
at the same
result
in the f o l l o w i n g
manner.
Let the f u n c t i o n
The
integral
f
be c o n t i n u o u s
(7) can then be w r i t t e n
on the
interval
is
(c,x).
as the sum of two i n t e g r a l s :
14
c D-v
x f(x) - F1(v) ~cX If (t)
(14)
f(x)] (x-t) v- ldt
-X
i
f(x)
r (v)
The first
tend to zero as
intervals
integral
v
F(v)
f(t)
1
r(v)
6
tegrals
on the right
tends to zero.
f(x)]
is any small positive
number.
in integral
B, where
is continuous.
these
in-
s
depends
1
6+0
After evaluating
is written
If(t)
on
6, and
f(x) I
by
s.
Thus,
lim s(6)
6+0
= 0
because
~xX
(x_t)V-ld t
6
~v
~ (~)~v
s(~)''T-:
r(v+l)
with
IBI
: o.
the second
integral
as
cDVf(x)-~_
IBI ÷ 0
of
v ~ O, we have
lim
(14)
B.
We then have
--< ~
where
Let us designate
f(x) l _-< ~(s)
1
IBI =< r--'~) ~(6)
for all
+
the maximum
If(t)
Eq.
(x-t) V-ldt
as
In B denote
Then,
into two sub-
~xX6 If(t)- f(x)] (x-t)V-ldt
A
f
in (14) will be shown to
It can be divided
of integration:
(15)
where
(x_t) V-ldt.
:
6.
= A ÷ B + (x - c ) v
r (v+l)
{(x)
on the right
in (14),
15
We can now consider
maximum of
If(t)
f(x) I
the integral
by M.
A
in (15).
Denote
the
Then
M
~c x-6 (x-t)V-ldt
IAI ~ ~7-~)
r(v+l)
(~v _ (x_c)V).
Let e be any arbitrary positive number.
IBI < e for all v ~ O. For this fixed
both sides of (16) add
-f(x).
Ken
cDxVf(x)
Because
f(x)
-i
.
IBI < e, we have
- f(x) I
~ 0 + ~ + 0
can be chosen as small as we wish,
lim I cDx vf(x)
v÷O
or
so that
v + O. To
$ [A I + IBI + If(x) l LF(v+l)
lim sup I cDx-v f(x)
v÷O
Since
Now choose
6
~, IAI + 0 as
lim
v÷O
D-Vf(x)
it follows
that
f(x) I = O,
= f(x).
C x
Another
approach
to the above result using the theory of
Laplace transforms might be of interest.
If we define
in [O,L] then f can be taken as zero in x > L. Let
f(x)
f(x)
only
be such
that
L
~ e_aXf (x) dx
exists
for some real
~(s)
is an analytic
a.
Then it follows
f
that
e-Stf(t)dt
function of
s
in
Re(s)
> a, that, with
x
e
-sx
1
r (v)
J
(x_t)v-lf(t) d t
v > -i,
16
exists in
Re(s) > a, and in fact, that
oo
7(s,v)
where
g(x,v)
=
-'o
e-SXg(x,v)
dx ~ s - v ~ ( s ) ,
denotes the right side of (7).
It is also true that
1 ~
f l (x) ="~T~
where
G
eS x ~(s) ds
is any vertical path lying in
differs from
f(x)
Re (s) > 0
and where
fl(x)
on, at most, a countable number of points.
Furthermore,
g(x,v) - 2 1~
But, for such a path
~G
e sx g(s,v) ds.
G, everything is uniformly bounded and
lim g(x,v) - 2 1~
v÷O
;
e st f(s)ds
z
/c
fl(x)
which is the result wantedj[16].
D-u CX
D-v f(x) = cDxu-v f ( x ) .
CX
We now consider criterion 5:
By definition
(17)
(7) we have
=
1
~c X (x-s) U-ld s
r(u)
1
F(vi
i
o
s
(S
_t)v- 1
f(t)dt.
The repeated integral above corresponds to a double integral to which
Dirichlet's formula, mentioned earlier, may be applied.
(18)
I
cDx u c D x-v f(x) - r(u)r(v)
/cx
We have
f(t) dt •
~-x
I
(x-s)U-I (s-t) v-I ds.
t
17
When either
to (18)
u
or
v
is on the interval
(0,I),
the passage
can be justified by a minor m o d i f i c a t i o n
from
(17)
of the Dirichlet
proof over a smaller triangle.
Make the transformation
gral on the right
in (18)
is a beta integral
The second inte-
is then
(x-t) u+v-I
which
y = (s-t)/(x-t).
fo1
(I -y)U-lyV-ldy
that has the value
r (u) r (v) (x_t)u+v-1
r (u+v)
When this
is substituted
into
(18), we obtain
~D~ u cDx ~ f(x)
= r(u+v)
~
1
The integral
on the right
the role of arbitrary
above is definition
order.
We then have
A subtle mathematical
the law of indices
order.
(7) with
u+v
playing
the required result.
arises when one seeks to extend
of arbitrary order to deriv-
If we follow the preceding method,
we will
integral
Du
Vf
=
1
c x cDx (x)
F(-u-v)
To establish
(19)
f
(x-t)
-(u+v)
-if(t)dt.
the relation
f(x)
c Du
x cD~f(x ) = c _u+v
Ux
it will be required
which vanishes
This proof
to impose the restriction
is omitted here but details
derivatives,
interchange
that
at the lower limit of integration,
The restriction
ing (19).
problem
stated for integration
atives of arbitrary
get the divergent
X(x_t)u+v_if (t)dt.
that
f
as stated in criterion
at
x = c
2, is necessary
f(c)
= O.
and at its n-i
to justify the
used in the proof of establish-
the relation
DD -I f(x) = DOf(x)
be a function
can be found in [6].
vanishes
of the order of operations
For example,
f
namely
= f(x)
18
always holds.
But the relation
(20)
is not
D-1D f ( x )
always valid.
=
DOf(x) = f ( x )
For, by definition
C D X-I cDx f(x)
= cDx- 1 f'(x)
= r(1)
=
and (20) holds only when
However,
sion.
by means
Let
greater than
-
(7) is for integration
of a simple
where
v, and
f(c),
= O.
of arbitrary
v = m-p
(x-t)Of'(t)dt
f(x)
f(c)
The definition
For differentiation
(7)
of arbitrary
order.
order it cannot be used directly.
trick, we can find a convergent
for convenience
O < p __< i.
m
is the least
Then for differentiation
expres-
integer
of arbi-
trary order we have
D v fix)
C
=
X
Dm
C
(21)
X
D -p f(x]
C
X
_ dm
1
dx m r(p)
~c x
(x-t)P-lf(t) dr'
where we take a~Ivantage of the knowledge
mth derivative
definition
operator
that
dm/dx m.
from the fact that
fractional
operator
quired analyticity,
by hindsight.
Dm
X
is an ordinary
We have assumed for purposes
referred
Dm-p
D -v.
to above, namely
is the analytic
It is obvious
of this
and also the other four criteria,
The question of extending
answered by letting
¢(v,x)
D v = Dm-p
continuation
that criterion
v
be real and greater
= oDxVf(x)
= F(v)
1
re-
of the
1 which re-
were established
the definition
gration of arbitrary order to differentiation
(22)
C
Dm-p = DmD -p.
The simple trick
sults
that
(7) for inte-
of arbitrary
than zero.
order
is
We have
~x
(x-t)v-lf(t)dt
~O
which
is in general
convergent
for
v > O.
For any
v
we can write
19
¢(v,x)
= oD-Vx f(x) = oDmx oDxp- f(x)
dm
1
dx m F(p)
where
-v = m-p,
When
m = O, I, 2,
v > O choose
~0 X
if
(x-t) p-
(t) dt,
....
m = O.
Thus
v = p
and
[~(v)~oX(x-t)v-lf(t)dtl
dx.
~ -- ,.
Now~
(22) can be written
~(v,x)
By Dirichlet's
= d~oX
formula, we have
d
1
~0 x (x-t)vf(t)dt
= ~-~ r(v+l)
~(v,x)
which
is convergent
~(v,x)
This process
point
of
v > -i.
= ~(V,X)
for
can be repeated
is analytic
v > -n.
for
Since
in
R1
v ~ -n, n
v > O
and
~
on a set of points
in the right half plane,
~.
m = I.
for
where
~ = ,
We then have
This justifies
then
q
in
Some explicit
examples
For the fractional
letting
v = m-p, m
RI~R 2
Dm-p
of fractional
derivative
for
in
R2
Now
for
continuation
Dv.
derivatives
of a constant
the least integer
integer.
with a limit
is the analytic
the trick of writing
useful.
a positive
is analytic
will be
k, we have by
> v, and the use of (21), the
formula
k
(23)
oD~, k - r ( I - v )
Another example
(24)
-v
is the integration
arbitrary order of the natural
By definition
x
and differentiation
logarithm.
(7) we have
O xD -v In x =
1/oX
F(v)
(x-t)V-lln
t dt,
V > O.
of
20
Let
t : x+
t - x,
: x(1
x>O
+ t~x).
in t = In x + in(l + t;x)
Then
with the restriction
-I
<
t-x < i .
X
Using the Taylor's
=
series
expansion
for
In(l+@),
we get
co
In t = in x +
~
(-l)n-l(t-x)n
L_.
nx
n
n=l
where the interval
of convergence
right side of the above
in x
is
O < t ~ 2x.
into the right
Substituting
the
side of (24) gives
~0 X (x_ t) v- Idt
r (v)
co
1
r(v)
Term by term integration,
gives
f
permissible
xVln x
In terms
because
xv
r(v)
XV
oDx v in x =
is Euler's
In x
~
of the psi function,
written
order of
dt
•
of uniform convergence,
co
in x = r(v+l)
C
(x-t) n
-- nx n
n=l
the result
oD;V
where
2
v-1
(x-t)
r (v+l)
constant.
[ In
x
-
1
~<~+k~
the above result can be
C
-
~(v+l)
]
For differentiation
of arbitrary
we have
O xD v I n x =
O xD m-p in x
am
dx m
(p+l)
xpZ
r(p)
k(
+
k=l
where the usual
criteria
for termwise
differentiation
is to be applied.
21
Although we now know how to interpolate
orders
where
of the derivative
such procedures
of functions
such as
might be applicable.
between
integral
in x, little
is known
In this connection
this
writer
s u b m i t t e d a p r o b l e m t o t h e American Mathematical Monthly t o
r(x).
This will permit
appear in winter 1974-75, concerning
0 Dxv i n
interpolation
b e t w e e n i n t e g r a l o r d e r s o f t h e p s i f u n c t i o n and m i g h t
h a v e u s e in t h e s u m m a t i o n o f s e r i e s o f t h e f o r m ~ 1 / ( 1 + x ) u.
Eric Russell
Love
[17] has defined
integration
of pure
imaginary order in such a way as to extend the properties
tion and differentiation
the case where
Re(n)
the Riemann-Liouville
cosine
of arbitrary
= O.
n
where
of integra-
Re(n)
Francis H. Northover makes
definition
(7) can be connected
and Fourier sine transforms
imaginary
order
> O
to
the claim that
to the Fourier
by means of derivatives
of pure
order as follows.
l~ X
1
cDxv- F(x)
I (x-t)V-IF(t)dt,
;cl
- r(v)
Re(v)
> O.
Make the transformation
t = x - (x-c)e -@.
The limits
(terminals)
of integration
then become
O
and
~, and we
have
cDxv- F(x)
v F(x
Now let
=
(X-c),V"
r(v)
-c)V
.... ( X
r(v$---S{
v = -in, and assume
e
d~),
-V~ F(#)
F
cD1xn F(x)
exists.
Then
oo
(26)
cDlxn F(x)
- (x-c)-{n
r(-in)
(x-c) -in
r(-in)
where
and
C(n) = (2)½
S(n)
= (2)½
50
L
e ~n~ F(~)
d~
½
(2)
{c(n)
F(~)
cos n~ d~
V(~) s i n
+ { s(n)}
n~ d~.
,
22
Love has shown that suitably
derivatives
of all orders
of any order
v
function which
for
Re(v)
where
> O.
restricted
Re(v)
functions
have
but have no derivative
but does not possess
For this reason,
paragraph where
= O
He has also cited an example
is locally integrable
of any imaginary order.
preceding
v
it was stated
of a
a derivative
caution was exercised
that
cD~ ~ F(x)
in the
is assumed
to
exist.
Consider now
.X
aDx- v
I =
Assume
f(t)
f(x) = F~v)
is expansible
/a
(x-t)v-lf(t)dt"
in a Taylor's
series
co
f(t) = 1
(-1)n f(n)(X)n! (x-t)n
n=O
The substitution
of the series
for
f(t)
in the integrand
above gives
co
1
(26a)
I
=
~
(-1) n f(n)(x ) (x-a) v+n
(v+n) n!
V (V)
n=O
Now if
f(x)
= (x-a) p,
p > -i, then
= F(p+v+l)
where we have noted without proof
r (v) ~ =
'r ( p + v + l )
If
=
_
the identity
1
( v + l ) r (p)
1
vr(p+l)
f(x) = (x-b) p ,
(x-a)V+P,
r(p+l)
_
aDx v(x-a)p
p > -i, then from
(x_a)V(x_b)p
aDxV(x-b)P
1
+
(v+2) 2!r
(26a)
y(_l)ncx_a~
F(v)
(p-l)
~x-b j
r(p+l)
......
(v+n) n! r(p-n+l)
n=O
forO<b~8.
We recall
D -u-v f(x)
the laws of exponents
is written
or indices
a D x-v a D x-u f(x) =
for the case when both terminals
of integra-
a x
tion are the s ~ e .
a measure
With
of deviation
the results
just given one can investigate
of the index rule,
say for example,
23
f(x) = x:
(See Open Questions,
aDx v bDx u x.
Some special
arbitrary
functions
oD;(P+½)
For
Re(p)
function.
376this
text).
as an integral
of
We wish to show the con-
function:
cos¢~¢~ = 2p /7 up-Zjo(/~).
> -½ , we have
Jp(X) =
(x/2) p
¢7 P (P+½)
Make the t r a n s f o r m a t i o n
Jp(X)
Let
can be represented
order of an elementary
nection with the Bessel
# 3, p.
=
(l-t2) p acos xt dt. [18]
xt = w, the above becomes
2
(2x) P/7 r (p+½)
SO x (x2-w2)p-½cos
w dw.
x 2 = u, w 2 = v, and the above becomes
~U
2pfg uP/2jp(¢~) = F(p+½)l *)Oi (u-v) p-½ c°scrv¢~ dv.
These transformations
forms
COS ~/~
/g
to our definition
have
given us an integral which
(7), of arbitrary
So, the above may be written
2p~-~ uP/2jp(~-~)
= oD~(P+½)
order
p+½,
and
con -
f(u) =
in the form
cos/~
¢g
which
is the result we sought to verify.
Here we show how a hypergeometric
sented by the fractional
operation
function
of a product
can be repre-
of elementary
func-
tions.
(27)
1 + lab
~ g x + a ( a + l ).b ( .b + l.) .x 2 . +
2!g(g+l)
is called a h y p e r g e o m e t r i c
the geometric
series
series because
1 + x + x 2 -'-
it is a generalization
The following notations
in common use:
(r)n = (r+l) (r+2)'''(r+n-l),
2Fl(a,b ;g;x) •
of
are
24
The subscript 2 preceding F denotes two parameters in the
numerator. The subscript 1 denotes one parameter in the denominator.
Using this notation, (27) can conveniently be written in summation
form:
(28)
2Fl(a,b;g;x)
(a)n(b)n n
n!(g)n x .
=
n=O
Some properties of the gamma and beta functions which will
be needed later are briefly outlined.
(b)n = b(b+l)'''(b+n-l)
(29)
(g)n
g(g+l)...(g+n-1)
Using the gamma-beta relation
(30)
Thus,
(b)n
~=
r(b+n)
=~ F(b)
r(~)
"r(g+n)
B(p,q) = F(p)F(q)/F(p+q),
(29) becomes
B(b+n. g-b]
B(b, g-b)
(28) becomes
co
~
1
2FI ta'b;g;xJ'' - B(b, g-b)
(31)
(a)/~B (b+n,g-b)
.x n
n!
n-O
where the factor I/[B(b,g-b)]
is placed before the summation sign
because it is independent of n.
Writing
symbol
2FI
B(b+n, g-b)
instead of
as a beta integral,
2Fl(a,b;g;x),
and using the
we then have
co
(32)
1
2F1 = B ( b , g - b )
r
~
/__z n!
xn
/oI(l_t) g_b_itb+n_id t "
n=O
The interchange of the summation sign and the integral sign
is permissible because of the uniform convergence of the series:
oo
(33)
1
2FI = B(b~g-b)
;oI(l-t)
g_b_itb- 1
(a)n(Xt) n
ni
=
Using the fact that
dr.
25
n•=O
(©n
n!
(xt)n = (1-xt)-a'
we find that
(33) becomes
1
2FI = ~
valid
if
(l-t)g
Ixl < i, and
All
the right
let
the integral
of the form of the definition
on
(7).
xt = s, and we have
x_g+l
the relation
gral above
adt,
that is required now is to transform
~x
2FI = B(b,g-b)
Using
-
-l(l-xt)
g,b > O.
above to an integral
To do this
-b -Itb
(x-s) g-b- isb-i (l-s) -ads .
10
B(b,g-b)
in operator
= r(b)r(g-b)/r(g),
notation,
we obtain
and writing
the inte-
the result
xg-lr (b)
-(g-b)x(b-l(l_x )-a
F(g)
2Fl(a'b ;g ;x) = oDx
Before
tional
calculus,
fractional
There
turning
it will be useful
integration
appears
our attention
to some applications
to mention
and another
another
definition
access to a fractional
to be two representations of Hermann
of frac-
Weyl's
of
derivative.
definition.
One is
f(x) = F(v)
1 rjJx~
x W-v~
The significant
Liouville
function
W ~+B
differences
definition
here being
for all
~
(t-x) v-l.
and
L
B •
definition
S. Miller
differential
integral
derives
operator
d n-I
+ Pl (x)
dx n-
1 + "'" + Pn (x)~
> O.
and the Riemann-
of integration
When the Weyl
Kenneth
be the linear
dx n
this
dt, Re(v)
way.
dn
L = Po(X)
between
are the terminals
gral in the following
Let
(t-x) v-I f(t)
and the kernel
exists,
the Weyl
W~W B =
inte-
26
whose
coefficients
finite
interval
adjoint
if
f
of
L
Pk' 0 ~ k ~ n, are of class
I = [a,b]
and
and
H (x,~)
Po(X)
its one-sided
is any function continuous
then for all
> 0
on
C~
on I.
Green's
I, and
xo
on some closed
Let
L
be the
function.
Then
is any point
in
I,
x ~ I,
(36)
g(x)
=
H (x,~)f(~)d~
Xo
is the solution
satisfies
of the nonhomogeneous
the boundary
g(k)(
Now let
xo = b
cited on p.90
.)
that
function
H (x,~)
for
Then if we let
g(x)
=
L.
=-H(~,x)
where H(x,~)
(See p. 37 of Miller's
is
text
xo = b
t t ( ~ , x ) f ( ~ ) d~
*
is the solution of
which
0 =< k =< n-l.
and recall
the one-sided Green's
L y = f(x)
conditions
) = O,
X O
equation
L y = f(x)
with initial
g(k)(b)
= O,
We recall
that
conditions
0 < k < n-l.
Now if
g
d n
-
~
~
dx n
then
L
is formally
self-adjoint
for this p a r t i c u l a r
L,
since
L
= (-l)nL.
(as in (i0)),
H(x,~)
i
- (n-l)!
g(x)
r(n)
(x_~)n-I
Thus
1
-
is the unique solution
(-i) n
~xb (~_x)n-lf(~) d ~
of the adjoint
equation
dnX,' = f(x)
dx n
(with the initial
may call
conditions
g(k) (b) = O,
0 =< k < n - 1 . )
So we
27
if
xWbv f ( x ) - F(w)
(g-x)W-lf(g)dg
the adjoint fractional integral
it)
,
Re
v
>
0
,
(unless someone else has already named
.
Now for
x
fixed,
lim
sufficient
a
condition
that
xWb~f(x)
b÷~
exists
is
f(x)
,
= 0
x < 0
and
foo
j
<
ix,21zfZ(x)d x
0
(Apply the
Cauchy-Schwarz
inequality.)
Formally
dx x w~'~f(x) = _x%(,~- 1 ) f(x)
and,
for e x a m p l e ,
xW~ (,J
Make
- 1)
e x
=
-~
r-~)
x = y
the t r a n s f o r m a t i o n
and we have
d~-~ ~,(,;)
e -x
xW-(V~ -I) e -x
d
(¢ x)'~ le-~d~'
e -x
yU_le -Ydy
r(,~)
-X
=
One n o t e s
e
that
-d m
w -v f(x)
dxmX~
= ( i] m w m - v
---x~
so that
W-½ e-X ~ e-X,
x ~
f(x)
X > O.
28
and
wm-½
x co
for any nonnegative
e-X
integer
= e-X
m.
The laws of exponents hold for
The argument
_
Re ~ > O
and
Re v > O.
is similar to (17) and (18):
_
,
xW'~[x W~vf(x)]
- F(~)F(~)
(t-x)~-idt
= r (~)r(~)
f(<)d<
fx
(~-t)~-if(~)d~
(t-x)~-l(~-t)W-ldt
oo
B(~ ,v)
=
~x f(~) (~-x) ~+V-ld~
X w-(~+v)f(x)
co
,
which is a law of exponents.
See also the paper by Kenneth S. Miller,
this text, pp. 80 - 90.
Another gorm of Weyl's
series
Mikol&s
involving
periodic
definition
in his paper this afternoon.
by A. Zygmund in the treatment
These definitions
of certain
One of the most recent methods
fractional
tend but their paper appears
of Butzer and Westphal,
function
of defining
~f(t)
the derivative
= ~(-l)J(~)
derivatives
different
are, unfortunately,
later in this text.
xY/F(y+l)
of arbitrary
is obtained
fy
(x) =
I
XY _
of
qnotient.
unable
to at-
Using the notation
order
as follows.
f(x-tj)
j=O
~
have been used
Fourier series.
order is by the limit of a fractional
Paul L. Butzer and Ursula Westphal
nonperiodic
is that of an infinite
functions which will be mentioned by Mikl6s
(x > O)
r(O+l)
(x < O)
~
of the
We define
29
[t-~&~f (x)]
t y
(s)
= t -a 1--~
sY+l
_
1
sy-a+l
(1-e-St)
~l-e-St) ~
st
=~[(f~-a(u) ,1g pa(~))(x)](s)
where the function
pa(x)
pa(x)
- r(~)
is defined by
(-1)J
(]) (x-j)a-1
X > O.
O__<j<x
pa(x)
belongs
to
Thus we h a v e
LI(o,~)
t - a A at f y ( x )
The f o l l o w i n g
the purposes
is
admittedly
kernel
consideration
equation
is
for
an opening
transform
a bit
s-a(i-e-S)
function
contrived,
of the
form
in the physical
it
is not
a
by fractional
serves
(x-t) v
sciences.
too far
a n d money f l o w i n s t e a d
solved
The problem is to determine
notch,
Laplace
t ÷ O+
formulated
of economics,demand
The i n t e g r a l
for
problem,
equation
deserves
a backdrop
+ fy-a
o f s h o w i n g how t h e
a n d an i n t e g r a l
frequency
and has the
fetched
of fluid
Its
to use
flow.
operations.
the shape
f(y)
of a weir
in a dam, in which the volume flow rate of fluid,
Q, through the notch is expressed
the notch~ [19]. We first establish
Q(h)
Consider
= c
as a function
h
of
(h-y)½f(y)dy.
front and side views
y
of the height
the equation
of the notch below:
Y
JlJiiiiJlJlJiiiJiiJJ
!
,
Th
Side view Fig. 1
Front view Fig.
2
30
Assuming
the points
a
(37)
y
in Fig.
equation
I, [20],we
can be applied between
obtain
Pa
V2
V2
- - + gh + a = Py + gY + _y_
p
2
p
2 '
where
and
that Bernoulli's
and
Pa' Py' Va' Vy
y; g
are the pressures
is the gravitational
and velocities
acceleration
and
p
at points
is the fluid
density.
The pressures
atmospheric
so that
be negligible
ing.
Thus,
at
a
and
Pa = Py'
(V a = O) since
y
are both
and the velocity
the fluid behind
taken to be nearly
at
a
is assumed
the notch
to
is slow mov-
(37) becomes
o
(38)
gh = gy + V~/2,
so that
(59)
gives
V
the velocity
Y
= (2g) ½(h-y) ½
y
of the fluid at distance
above
the notch
floor
(x-axis).
The elemental
(40)
Denoting
2(2g) ½
the elemental
by
c
volume
(42)
Q(h)
We find
(43)
region
in Fig.
2) is given by
volume
flow rate through
dA
is
to
y = h
dQ = V dA = 2 ( 2 g ) ½ ( h - y ) 2 f ( y ) d y .
Y
gives the total
fractional
(shaded
dA = 2f(y) dy.
So, by definition,
(41)
area
f(y)
integration
Q(h)
and integrating
flow rate through
= c
from
y = O
the notch:
(h-y) ½f(y)dy.
by finding
(7)~ Eq.
= r(~)
(41)
(42)
f(h).
By the definition
can be written
in the form
oDh 3/2 f(h).
C
Operating
(44)
on both sides of (43) with
f(h)
i
F(3/2)
oD~/2
oD~/2qc(h)
gives
of
the result
31
312
But
-
oDh
= O D2 oDh ½
g(h), we can write
2
where
(44)
oDh
is
d21dh 2
.
Denoting
Q(h)/c
by
as follows:
-h
1
M 2
1
fO (h-Y) -½g(y) dY"
f ( h ) - F(3/2) dh 2 F(%)
(45)
Since
Q(h)/c
i s known, then
after evaluating
second
the beta
derivative
for
and if
a > -½.
a = 3/2,
R.I.
lar symmetry
fluid exits
by
suggests
above
where
the weir
a similar
height
We will now
= O.
its
(45) yields
like a parabola,
are
h
at
and
H.
of circu-
and maximum heights
H
respectively.
of
The
The time for the fluid to
the ground,
Determine
Robert M. Hashway,
A fluid reservoir
The minimum
above
consider
for the nth derivative
Then
is shaped
problem.
an orifice
t(H)
= ha .
to this writer,
the ground
through
g(h)
and taking
f(y).
is a rectangle.
communication
a particular
t(z)
a = 7/2,
is to be designed.
the reservoir
reach
let
above,
we have
Then,
r (a-h)
the weir
In his
Warwick,
If
Thus,
are known.
r(a+l__.___~)
ha_3/2
7[ 2
valid
g(y)
on the right
f(h).
example
f(h) - 2
and
integral
we obtain
As a specific
g(h)
say height
the shape
a generalization
z, is given
of the reservoir.
of Leibnitz's
rule
of a product:
oo
oDVf(x) g(x)
(46)
=
~,
~(n)
(v) OVx
f(x)
oD(V-n) g (x) .
n=O
D (n)
is ordinary
entiation.
differentiation
Consider
and
D (v-n)
is fractional
differ-
the identity
x a+b = xax b
Operate
on both sides with
tional derivative
right hand side
like powers,
F(a+b+l)
r(a+b+v+l)
D v.
Treat
the left hand side as a frac-
in accord with the definition
in accord with
we get an infinite
= r(b+l)
I I
F(b+v+l)
(46).
series
(21)
By equating
and treat the
coefficients
of
of gamma functions:
va
F(b+v+2)
v(v+l)a(a -I)
+ 2!F(b+v+3)
I
"'"
"
32
The case for fractional
plicity
it offers
the Volterra
type.
(483
on b o t h
of
explicitly
for convenience
F(½)
D -½ f(x).
O x
we have
= ¢~ D "½ f ( x ) .
sides
D2xf(x)
of the
above with
D½
yields
= ¢~ f ( x ) .
(46) to get
(50)
xD½f(x)
Substituting
equations
f(x)
(x-t) -½ f(t)dt.
subscripts
xf(x)
Apply formula
=
(7) the right hand side above is
Omitting
(49)
integral
2x
xf(x)
Operating
might well lie in the sim-
Consider the problem of finding
given the equation
By definition
calculus
in the solution of certain
+ 1D-½f(x)
= ¢~ f ( x ) .
(48) into (50) gives
(51)
xD½f(x)
x f ( x ) ....
+ 2vr~
We can get an expression
for
= ¢~ f ( x ) .
D½f(x)
by operating
on both sides of
(48) with D:
(52)
D[xf(x)]
= ¢~ D½ f ( x ) ,
or
(53)
xf'(x)
Our objective
+ f(x) = ~
D ½ f(x).
has been reached when
We arrive at the ordinary
x2f,(x)
(53) is substituted
differential
into
(51).
equation
+ ( -3yx _ 7 ) f(x) = 0
which has the solution
f(x) = ke-~/Xx -3/2,
Murray R. Spiegel,
texts in the Schaum's outline
to the previous
problem.
author of Laplace Transforms and other
series,
suggests
the following
solution
33
xF(x)
=
_ d_
~0X( x - u ) - ½ F ( u ) d u
= x -½ * F ( x ) .
f(s)
= ~
= £(½---~) f ( s )
ds
-
s½
f'(s)/f(s)
f(s)
s~
'
= / ' ~ s -½
In f(s) = 2/-~ s ½ + c I
f(s) = ce -2/-~
F(x) = c X -I
e
- -
e
-4~/4x
2 ~JTTT~
= cx-3/2e-~/x
No claim can be made that the fractional
is better than some other approach.
Parker Higgins who confided
However,
calculus
approach
to paraphrase
Theodore
in me tha~ he paraphrased
there is a succinctness
of notation
the fractional
that might suggest a solution
functional
calculus
equation
that is not readily
In 1940 and 1941 Erd&lyi
of a generalization
tions.
Professor
calculus
sixties,
lay relatively
Higgins,
techniques
dormant
for the solution,
began.
AI-Bassam,
Of particular
More papers
Osler and
interest
by means of fractional
equations
of mathematical
to the
of some
operations,
that stem from mixed boundary
physics.
The pair of equations
f O ~ K(x,t)G(t)f(t)dt
The
from 1941 to the
in the last decade was the development
of dual and triple integral
value problems
properties
and of the Weyl defini-
resurgence
Mikol~s,
in the 1960's and early 1970's.
applied mathematician
formal
and Kober investigated
when a modest
in
to a complicated
obtained by other means.
of the Riemann-Liouville
were published by Erd~lyi,
others
of formulation
Sneddon will survey some of these results.
topic of fractional
early nineteen
and simplicity
A. Erd&lyi,
= g(x)
O<x<l,
34
O~ K ( x , t ) f ( t ) d t
where the
f(t)
is
kernel
K(x,t),
G(t),
t o be d e t e r m i n e d
to
are
a specific
g(x),
is
gral
equations,
tial
in the field of an electrified
for
ient to determine
gral equations.
problem
finding
known f u n c t i o n s
integral
to
an e x p r e s s i o n
of fractional
fill this void.
for
calculus
it is often conven-
is to popularize
to include
ricula.
is to exchange
Another purpose
in the
in his lectures.
Many matheThe wide
will help to
singular purposes,
One obvious
the topic in the hope it will induce
tists and mathematicians
inte-
further trends
at this conference
has several
hold
a set of triple
singular being taken in the sense of Sherlock Holmes.
purpose
the poten-
conditions
are unfamiliar with this topic.
This conference
inte-
boundary
is old but studied little.
to be presented
The
of dual
parts of the same boundary.
parts of the same boundary,
and scientists
of papers
a pair
is such that different
calculus
and
equations.
disc where different
Professor Mikol~s will discuss
Fractional
variety
are
dual
the solution by constructing
theory and applications
maticians
in
hold over two different
over three different
h(x)
physical
example,
When the problem
x > 1
known as
idea
conditions
reduce
= h(x)
it in their research
scien-
and cur-
and impart information
which
may serve to suggest new areas of research.
Fractional
matics.
proper objects
in the last decade,
in various
chemistry,
can be categorized
fields:
scattering
rheology,
theory,
theory and elasticity.
and scientists
are unfamiliar
However,
another objective
with mathematical
fractional
calculus
electro-
theory, probabil-
many mathematicians
with this topic possibly because
calculus has developed,
of this conference
formal methods
they
Thus, while the theory
its use has lagged behind.
is to encourage
of representing
attempts
physical
So,
to dis-
phenomena
models that can be treated with the elegance
calculus.
are
and applied
biology,
transport
have not been exposed to its applications.
cover additional
Scientists
quantitative
diffusion,
mathe-
operators
found the fractional
ity, potential
of fractional
as applicable
and theory of these fractional
of study in their own right.
mathematicians,
useful
calculus
The properties
of
35
REFERENCES
[I] Leibnitz, G.W., Leibnitzen's Mathematische Schriften,
Germany: Georg Olm, 1962, v. 2, pp. 301-302.
Hildesheim,
[2] Lacroix, S.F., Trait~ du Calcul DiffJrentiel et du Calcul Integral, Paris: Mme. vecourcier, 1819, Tome Troisi&me, seconde &dition,
pp. 409-410.
[3] Spanier, Jerome and Oldham, Keith B., The Fractional Calculus,
New York: Academic Press, 1974.
[4] Abel, Niels Henrik, "Solution de quelques probl~mes a'l'aide
d'int&grales d~finies," Oeuvres Completes, Christiania, 1881, tome
premiere, 16-18.
[5] Liouville, Joseph, "M&moire sur quelques Qu&stions de G&ometrie
et de M&canique, et sur un nouveau genre de Calcul pour r&soudre ces
Qu&stions," Journal de l'Ecole Polytechnique, 1832, tome XIII, XXI e
cahier, pp. 1-69.
[6] A more detailed discussion of Liouville's first and second definitions and also of their connection with the Riemann definition can
be found in The Development of the Gamma Function and A Profile of
Fractional Calculus, by Bertram Ross, New York University dissertation, 1974, Chapter V, pp. 142-210.
University Microfilms, Ann Arbor,
Mich., #74-17154, PO #45122.
[7] Debnath, Lokenath and Speight, T.B., "On Generalized Derivatives,"
Pi Mu Epsilon Journal, v. 5, 1971, ND 5, pp. 217-220, East Carolina
University.
[8~through[ll] Details will be found in "A Chronological Bibliography
of Fractional Calculus with Commentary," by Bertram Ross in The Fractional Calculus [3], pp. 3-15, and in [6].
[12] Davis, Harold Thayer, The Theory of Linear Operators,
ton, Indiana: The Principia Press, 1936; p. 20.
[13]
Blooming-
See [6], pp. 158-162.
[14] The first one to apply Dirichlet's method to kernels of the form
(x-t) v is Wallie Abraham Hurwitz in 1908.
Cited by Whittaker and
Watson, A Course in Modern Analysis, 4th edition, 1963, p. 76.
[15] I am indebted to Dr. Kenneth S. Miller, Riverside Research
Institute, New York City, for this contribution.
[16] This approach was recommended by George F. Carrier, Harvard
University.
[17] Love, Eric Russell, "Fractional Derivatives of Imaginary Order,"
The Journal of the London Mathematical Society, Volume III (Second
Series), 1971, pp. 241-259.
[18] Farrell, Orin J. and Ross, Bertram, Solved Problems in Analysis,
New York: Dover Publications, 1971, 279. First published in 1963,
New York: The Macmillan Co.
36
[19] Brenke, W.C., "An Application of Abel's Integral Equation,"
American Mathematical Monthly, 1922, v. 29, 58-60.
[20] Bernoulli's equation is strictly valid for steady, frictionless
flow in a stream tube.
It is used, however, in engineering for flows
with friction by modification of solutions with a suitable friction
factor.