MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 12 doc

OTHER DEFINITIONS OF DIFFERINTEGRALS
393
z-phe
t
Fig.
14.3
Contour
C'
=
C
+
Co
+
LI
+
Lz
in
the
differintegral formula
which goes to zero in the limit
60
+
0.
For
the
CO
integral to be zero in the
limit
6,
4
0,

we have taken
q
as
negative. Using this result we can write
Equation (14.74)
as
Now we have to evaluate the
[
f+L,
-
f
+Lz
3
integral.
we
first evaluate the
parts
of
the integral
for
[-m,
01,
which gives zero
as
=
0.
394
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFfRlNTEGRALS"
Fig.
14.4

Contours
for
the
$+L,,
$+,,,
,
and
jC
integrals
Writing the remaining part
of
the
$,
dz
integral we get
(:!$I
(14.82)
After taking the limit we substitute this into the definition
[Eq.
(14.74)] to
obtain
(14.83)
Simplifying this we can write
(14.84)
(14.85)
To
see
that this agrees with the Riemarin-Liouville definition we
use
the

fol-
lowing relation of the gamma function:
and write
(14.86)

dqf(x)
'('
Ids
q
<
0
and noninteger. (14.87)
dxq
I
r(-q)
(x-6)4+"
OTHER DEF/N/T/ONS OF DIFFERINTEGRALS
395
This is nothing but the Riemann-Liouville definition. Using Equation
(14.71)
we can extend this definition to positive values of
q.
14.3.2
Riemann
Formula
We now evaluate the differintegral
of
f
(x)
=

xp,
(14.88)
which
is
very useful
for
finding differintegrals of functions the Taylor series
of
which can
be
given. Using formula
(14.84)
we write
SP
d6

(14.89)
dxq
?I-
s
0
(6-x)4+1
dqxp

r(q
+
l)
sin(?I-q)(-1)9
and
We define

-
-s
6
-
X
so
that Equation
(14.90)
becomes
Remembering the definition of the beta function:
we
can write Equation
(14.92)
as
Also using the relation
(14.86)
and
between the beta and the gamma functions, we obtain the result
as
dqxp
r(p
+
1)xP-q
dxq
p
>
-1
and
q
<

0.

-
r(p
+
1
-
q)
'
(14.90)
(14.91)
(14.92)
(14.93)
(14.94)
(14.95)
(14.96)
396
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
Limits on the parameters
p
and
q
follow from the conditions
of
convergence
for
the
beta
integral.
For

q
2
0,
as
in the Riemann-Liouville definition, we write
(14.97)
and choose the integer
n
as
q
-
n
<
0
.
We now evaluate the differintegral
inside the square brackets using formula (14.71)
as

(14.98)
Combining this with the results in Equations (14.96) and (14.98) we obtain
a
formula valid
for
all
q
as
dqxP
r(p
+

1)xP-q
-
p>-1.
dxq
r(p
-
q
+
1)
'
(14.99)
This formula is
also
known
as
the Riemann formula. It is
a
generalization
of
the formula
m!
-
xrn-",
6"X"
dxn
(m-n)!
(14.100)
for
p
>

-1,
where
m
and
n
are positive integers.
For
p
5
-1
the
beta
function
is divergent. Thus
a
generalization valid for all
p
values is yet
to
be
found.
14.3.3
Differintegrals via Laplace Transforms
For
the negative values
of
q
we can define differintegrals by using Laplace
transforms
as

dqf
-
-E-'[sq&]
,
q
<
0,
dxq
(14.101)
where
F(s)
is the Laplace transform
of
f(x).
To
see
that this agrees with the
Riemann-Liouville definition we make use
of
the convolution theorem
In this equation we take g(x)
as
(14.103)
OTHER DEFINITIONS
OF
DIFFERINTEGRALS
397
where its Laplace transform is
(14.104)

=
r( q)s*,
(14.105)
and also write the Laplace transform
of
f(x)
as
For
q
<
0
we obtain
["'I
=
-c-"sq&)]
,
q<o.
dxq
(14.106)
(14.107)
(14.108)
The subscripts
L
and
R-L
denote the method
used
in evaluating the differin-
tegral. Thus the two methods
agree

for
y
<
0.
For
q
>
0,
the differintegral definition by the Laplace transforms
is
given
as
(Section
14.6.1)
or
dq-lf
dxg-
sqF(~)
-
-(o)
-
. . .
-
sn-'-(O)]
dxq-n
.
(14.110)
In
this definition
q

>
0
and the integer
n
must
be
chosen such that the
inequality
n
-
1
<
q
5
n
is
satisfied. The differintegrals on the right-hand
side
are
all evaluated via the
L
method.
To
show that the methods
agree
we
write
and use the convolution theorem
to
find its Laplace transform

as
(14.112)
=
~q-~F(s),
q
-
n
<
0.
(14.113)
398
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
-
This gives us the
sqf(s)
=
snx(s)
relation.
definition
[Eqs.
(14.7O-71)] we can write
Using the Riemann-Liouville
Since
q
-
n
<
0
and because
of

Equation (14. 108)' we can write
From the definition
of
A(z)
we can
also
write
A(z)
=
-
q-n<O,
r(n
-
q)
Jx
,,
(z
-
f(r)dr
T)Q-"+~
'
(14.114)
(14.115)
(14.116)
As
in the Griinwald and Riemann-Liouville definitions we assume that the
[ ]L
definition also satisfies the relation
[Eq.
(14.40)]

(
14.117)
where
n
a
is positive integer and
q
takes all values. We can now write
which gives us
Similarly we find the other terms in Ekpation (14.110) to write
(14.118)
(14.119)
(14.120)
PROPERTIES
OF
DIFFERINTEGRALS
399
Using Equation (14.111) we can now write
(14.122)
which shows that for
q
>
0,
too, both definitions agree.
In formula (14.1 lo),
if
the function
f(x)
satisfies the boundary conditions
(14.123)

we can write a differintegral definition valid for all
q
values via the Laplace
transform
as
-=
dqf
-E-l[sQ~(s)].
dxq
(14.124)
However, because the boundary conditions (14.123) involve fractional deriva-
tives this will create problems in interpretation and application. (See Problem
14.7 on the Caputo definition
of
fractional derivatives.)
14.4
PROPERTIES OF DIFFERINTEGRALS
In this section
we
see
the basic properties of differintegals. These properties
are also useful in generating new differintegrals from the known ones.
14.4.1
Linearity
We express the linearity
of
differintegrals
as
d9f2
+

d9[fl
+
f2l
-
d4f1
-
[d(x
-
~)]q
[d(x
-
~)]q
[d(X
-
~)]q*
(14.125)
14.4.2
Homogeneity
Homogeneity
of
differintegrals
is
expressed
as
Co
is any constant. (14.126)
dQ(Cof)
=Co
dqf
[d(z

-
~)]q
[d(x
-
~)]q
'
Both
of
these properties could easily be seen from the Griinwald definition
[Eq. (14.39))
400
FRACTIONAL DERIVATIVES AND INTEGRALS. "DIFFERIN TEGRALS"
14.4.3 Scale Transformation
We express the scale transformation
of
a
function with respect to the lower
limit
a
as
f
+
f(rz
-
ya
+
4,
(14.127)
where
y

is
a
constant scale factor. If the lower limit is zero, this means that
f(4
-+
f(r.)-
(14.128)
If the lower limit differs from zero, the scale change is given
as
dqf(yX)
x
=
z
+
[a
-
ay]/y
(14.129)
dQf(yX)
-
-
[d(z
-
a)]q
"[d(yX
-
a)]Q'
This formula
is
most useful when

a
is zero:
(14.130)
14.4.4 Differintegral of
a
Series
Using the linearity
of
the differintegral operator we can find the differintegral
of
a
uniformly convergent
series
for
all
q
values
as
(14.131)
Differintegrated series
are
also uniformly convergent in the same interval.
For
functions with power series expansions, using the Riemann formula we can
write
dQ
c
O0
-
a]P+(j/n)

=
I-
(pn
+: +
[.
-
a]P-q+(.i/n)
00
(14.132)
where
q
can take any value, but
p
+
(j/n)
>
-1,
a0
#
0,
and
n
is
a
positive
integer.
)
[d(z
-
.)I"

j=o
zaj
I-
(
pn
-
q;+
j
+
n
14.4.5 Composition
of
Differintegrals
When working with differintegrals one always has
to
remember that operations
like
dqd&
=
dQdQ,
dQ&
=
dq+Q
and
d9f
=g+
f
=d-qg
(14.133)
PROPERTIES

OF
DIFFERINTEGRALS
401
are valid only under certain conditions. In these operations problems are not
just restricted
to
the noninteger values
of
q
and
Q.
When
n
and
N
are positive integer numbers, from the properties
of
deriva-
tives and integrals we can write
d"
dN
f
dn+N
f
[d(z
-
a)]"
{
[d(z
-

a,].}
=
[d(z
-
a)]n+N
(14.134)
-
-
and
f
(14.135)
d-n-N
-
However,
if
we look
at
the operation
[d(z
d*n
-
a)]*"
{
[d(~
diNf
-
.)ITN
1,
(14.136)
the result is not always

d*niN
f
[d(z
-
u)]*"?"
(
14.137)
Assume that the function
f
(z)
has continuous Nth-order derivative in the
interval
[a,
b]
and
let
us
take the integral
of
this Nth-order derivative
as
We integrate this once more:
and repeat the process
n
times
to
get
(14.140)
(.
-

a)"-'
1-
f
(N-
yu).
(n
-
l)!
402
FRACTIONAL DERIVATIVES AND INTEGRALS- "DIFFERINTEGRALS"
Since
we write
(14.142)
Writing Equation
(14.142)
for
N
=
0
gives
us
n-
1
k!
k=O
[d(z
-
a)]-"
(14.143)
We differentiate this to get

f("-")(a).
(14.144)
[x
-
a]k-
n-
1
(k
-
l)!
k=
1
After
N-fold
differentiation we obtain
f('"-")(u).
(14.145)
I.
-
n-
1
(k
-
N)!
k=N
For
N
2
n,
remembering that differentiation

does
not depend
on
the lower
limit and also observing that in this case the summation in Equation
(14.145)
is empty, we write
dN-nf
=
f(N-")(z).
(14.146)
On the other hand for
N
<
n,
we use Equation
(14.143)
to write
dN-n
n-N-1
(a).
(14.147)
k!
[d(z
-
f3)lN-n
k=O
This equation also contains Equation
(14.146).
In Equation

(14.145)
we now
make the transformation
k+k+N
(14.148)
to write
n-N-1
(a)-
k!
(14.149)
PROPERTIES
OF
DIFFERINTEGRALS
403
Because the right-hand sides of Equations
(14.149)
and
(14.147)
are
identical,
we obtain the composition rule for
n
successive integrations followed
by
N
differentiations
as
(14.150)
To
find the composition rule for the cases where the differentiations are

performed before the integrations, we turn to Equation
(14.142)
and write
the sum in two pieces as
Comparing this with Equation
(14.147),
we now obtain the composition rule
for the cases where N-fold differentiation is performed before
n
successive
integrations
as
n-
1
k
(a).
-
(N+k-n)
f
k!
k=n-
N
(14.152)
Example
14.1.
Composition
of
differintegrak:
For
the function

f (x)
=
,
we
first
calculate
e-3x
For
this case we use Equations
(14.150)
and
(14.143)
to find
d-3
f
(2)
d-2
f
(x)
&{F>=w
e-3x
XI
+
9
39
- -
(14.153)
On the other hand, for
we have to use formula
(14.152).

Since
N
=
1
and
n
=
3,
k
takes only
the value two, thus giving
(14.154)
404
FRACTIONAL DERIVATIVES AND INTEGRALS: “DIFFERINTEGRALS”
14.4.5.1
any value, composition
of
differintegrals
as
Composition Rule
for
General
q
and
Q:
When
q
and
Q
take

(14.155)
dq+Q
f
[d(z
dq
-
~)]q
[d(z
dQf
-
a)]&
1
=
[d(z
-
a)]q+Q
is
possible only under certain conditions. It is needless to say that we assume
all the required differintegrals exist. Assuming that
a
series expansion for
f
(z)
can be given
as
W
f
(z)
=
uj[z

-
aIp+j,
p
is
a
noninteger such that
p
+
j
>
-1, (14.156)
j=O
it can be shown that the composition rule [Eq.
functions satisfying the condition
(14.155)]
is valid only for
(14.157)
In general, for functions that can be expanded
as
Equation
(14.156)
differin-
tegrals are composed as (Oldham and Spanier)
- -
(14.158)
For
such functions violation
of
condition
(14.157)

can be shown to result from
the fact that
&
vanishes even though
f
(x)
is different from zero. From
here we see that, even though the operators
60
and
&a
are in
general inverses of each other, this is not always true.
In practice it
is
difficult to apply the composition rule
as
given in Equation
(14.158).
Because the violation
of
Equation
(14.157)
is
equivalent to the
dQf(x)
vanishing of the derivative
Q‘,
let us first write the differintegral (for
&I

simplicity we set
a
=
0)
of
f(z)
as
Because the condition
p
+
j
>
-1
(or
p
>
-1),
the gamma function in the
numerator is always different from zero and finite.
For
the
Q
<
p
+
1
values,
gamma function in the denominator is always finite; thus condition
(14.157)
is satisfied. For the remaining cases condition

(14.157)
is violat,&. We now
PROPERTIES
OF
DtFFERlNTEGRALS
405
check the equivalent condition
=
0
to identify the terms responsible
for the violation of condition
(14.157).
For
the derivative
to
vanish,
from Equation
(14.159)
it is seen that the gamma function in the denominator
must diverge for all
uj
#
0,
that is,
[dxl
p+j
-
Q
+
1

=
0,
-1,
-2,

.
For
a
given
p
(>
-1)
and positive
Q,
j
will eventually make
(p
-
Q
+
j
+
1)
positive; therefore we can write
p+j
=
Q
-
l,Q
-

2,
,
Q
-
m
(14.160)
where
m
is an integer satisfying
O<
Q<m<
Q+1. (14.161)
For
the
j
values that make
(p
-
Q
+
j
+
1)
positive, the gamma function
in the denominator is finite, and the corresponding terms in the series satisfy
condition
(14.157).
Thus the problem
is
located to the terms with the

j
values
satisfying Equation
(14.160).
Now, in general for an arbitrary diffferintegrable
function we can write the expression
d-Q
f
(z)
-
-
[
51
=
coxQ-'
+
clzQ-2
+
.
+
cmzQ-,,
(14.162)
[dzl-Q
[&]Q
where
c1,
c2,
)
c,
are arbitrary constants. Note that the right-hand side of

Equation
(14.162)
is
exactly composed
of
the terms that vanish when
#
0,
that is, when Equation
(14.157)
is
satisfied. This formula, which
IS
very
useful in finding solutions of extraordinary differential equations can now be
used in Equation
(14.158)
to compose differintegrals.
Another useful formula is obtained when
Q
takes integer values
N
in Equa-
tion
(14.158).
We apply the composition rule [Eq.
(14.158)]
with Equation
(14.142)
written for

n
=
N,
and use the generalization of the Riemann for-
mula:
1"11
(14.163)
to obtain
-
dq+N
f
- -
[d(z
-
U)]"+"
(14.164)
406
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
Example
14.2.
Composition
of
diffeerintegmls:
As
another exampie we
consider the function
f
=
x-w
(14.165)

for
the values
a
=
0,
Q
=
1/2, and
q
=
-1/2. Since condition (14.157)
is
not satisfied, that
is,
(14.166)
#
0,
(14.167)
- -
x-
1/2
-
0
we have
to
use Equation (14.158):
(14.168)
Since
we have
which

leads
to
(14.169)
(14.170)
(14.171)
1
Contrary
to
what we expect
d-'
is
not the inverse
of
dz
for
x-lI2.
a
3
Example
14.3.
Inverse
of
differintegmk:
We
now consider the function
PROPERTIES
OF
DIFFERINTEGRALS
407
f

=x
(14.174)
for the values
Q
=
2
and
a
=
0.
Since
d2
x
-=0
[d5Cl2
is
true, contrary to
our
expectations we find
d-'
d2x

=
0.
[d~]-~ [dxI2
(14.175)
(
14.176)
The problem
is

again that the function
f
=
x
does
not satisfy condition
(14.157).
14.4.6
Leibniz's Rule
The differintegral of the qth order of the multiplication of two functions
f
and
g
is given by the formula
where the binomial coefficients are to
be
calculated by replacing the factorials
with the corresponding gamma functions.
14.4.7
Right- and Left-Handed Differintegrals
The Riemann-Liouville definition of differintegral was given
as
where
k
is an integer satisfying
k=O
for
q
<
0

k-l<q<k
for
420.
(
14.179)
This
is
also
called
the right-handed Riemann-Liouville definition. If
f
(t)
is
a
function representing a dynamic process, in general
t
is
a
timelike variable.
The principle
of
causality justifies the usage of the right-handed derivative
because the present value of
a
differintegral
is
determined from the past values
off
(t)
starting from an initial time

t
=
a.
Similar to the advanced potentials,
it
is
also possible to define
a
left-handed Riemann-Liouville differintegral
as
408
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
where
k
is
again an integer satisfying Equation (14.179). Even though for
dynamic processes it is difficult to interpret the left-handed definition] in
general the boundary
or
the initial conditions determine which definition is
to
be
used.
It is
also
possible
to
give
a
left-handed version of the Griinwald

definition. In this chapter we confine ourselves to the right-handed definition.
14.4.8
We now discuss the dependence
of
Dependence on the
Lower
Limit
dq
f
fx)
-
.)I"
(14.181)
on the lower limit.
For
q
<
0,
using Equation (14.178) we write the difference
dQf(4
-
dQf
6=
[d(x
-
u)]Q
[d(.
-
b)]Q
as

(14.182)
For
the binomial coefficients we write
(
-"iq
)
=
to obtain
d-
(
see
Section 14.5.1)
03
-
-
I
=o
(14.184)
(14.185)
DIFFERINTEGRALS OF
SOME
FUNCTIONS
409
Even though we have obtained this expression for
q
<
0,
it is also valid for all
q
(Oldham and Spanier, Section

3.2).
For
q
=
0,1,2,
,
that
is,
for ordinary
derivatives, we have
S=0
(14.186)
as expected.
For
q
=
-1
the above equation simplifies to
(14.187)
For all other values of
q,
S
not only depends on
a
and
b
but also on
x.
This is
due

to
the fact that the differintegral, except when it reduces to an ordinary
derivative, is a global operator and requires
a
knowledge of
f
over the entire
space. This is apparent from the Riemann-Liouville definition
[Eq.
(14.178)],
which is given
as
an integral, and the Griinwald definition
[Eq.
(14.39)]
which
is given
as
an infinite series.
14.5 DIFFERINTEGRALS OF
SOME
FUNCTIONS
In this section we discuss differintegrals of some selected functions. For an
extensive list and discussion
of
the differintegrals of functions of mathematical
physics we refer the reader to Oldham and Spanier.
14.5.1 Differintegral
of
a

Constant
First we take the number one and find its differintegral using the Griinwald
definition [Eq.
(14.39)]
as
Using the properties of gamma functions;
Cy=il
r(j
-
q)/I'(-q)I'(j
+
I)
=
r(N
-
q)/r(l-
q)r(N),
and limN,oo[N*r(N
-
q)/r(N)]
=
1,
we find
(14.189)
When
q
takes integer values, this reduces to the expected result.
arbitrary constant
C,
including zero, the differintegral

is
(see
Problem
14.7)
For an
(14.190)
410
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
14.5.2
For
the differintegral
of
[z
-
a]
,
we again use Equation (14.39)
and
write
Differintegral
of
[Z
-
a]
q
N-l
dq[x
-
U]
r(j

-
q)
[,v,
-;
+ja
[d(z
-a)]"
N+cc
(14.191)
In addition to the properties
used
in Section 14.5.1, we
also
use the fol-
lowing relation between the gamma functions:
C,"='r(j
-
q)/r(-q)r(j)
=
(-q)r(N
-
q)/r(2
-
q)r(N
-
l), to obtain
dq[z
-
u]
-

[z
-
4-q
[d(z
-
u)]q
-
r(2
-
q)
'
(14.193)
(14.194)
We now use the Riemann-Liouville formula to find the same differintegral.
We first write
dq[~
-
U]
[z'
-
a]dz'
-
r(-d
[z
-
z']q+l
For
y
<
0

values we make the transformation
y
=
z
-
z'
and write
dQ[x
-
U]
[d(z
-
.>1q
rY-4
(14.195)
(14.196)
which
leads
us
to
(14.198)
(14.199)
(14.200)
DIFFERINTEGRALS
OF
SOME
FUNCTIONS
411
For
the other values of

q
we use formula
(14.40)
to write
1
(14.201)
and choose
n
such that
q
-
n
<
0
is
satisfied. We use the Riemann formula
[Eq.
(14.99)]
to write
(14.202)
which
leads
to the following result:
-
r(2
-
+
n)
[Z
-

-
r(2
-
q)
r(2
-
+
-
[x
-
up
-
4)
.
-
(14.204)
(14.205)
This is now valid for all
q.
14.5.3
Here there is no restriction on
p
other than
p
>
-1.
We start with the
Riemann-Liouville definition and write
Differintegral
of

[Z
-
u]”
(p
>
-1)
(14.206)
When we use the transformation
z’
-
a
=
v,
this becomes
Now
we make the transformation
v
=
(x
-
a)u
to write
Using the definition of the beta function [Eq.
(13.151)]
and its relation with
the gamma functions, we finally obtain
(14.209)
(14.210)
where
q

<
0
and
p
>
-1.
Actually, we could remove the restriction on
q
and
use Equation
(14.210)
for all
q
(see
the derivation of the Riemann formula
with the substitution
x
-+
x
-
a
).
412
FRACTIONAL DERIVATIVES AND
INTEGRAlSr"DIFFERINTEGRA1S"
14.5.4
To
find
a
formula valid

for
all
p
and
q
values we write
Differintegral
of
[1
-
z]~
1
-
z
=
1
-
u
-
(z
-
u)
and
use
the binomial formula to write
(14.211)
M
(1-
p-
r(pf

')
(-l)j(l
-
u)p-j(z
-
a)j.
(14.212)
-
r(j
+
i)r(p
-
j
+
i)
We now
use
Equation
(14.132)
and the Riemann formula
(14.99),
along with
the properties
of
gamma and the
beta
functions to find
where
B,
is

the incomplete
beta
function.
14.5.5
Differintegral
of
exp(fz)
We first write the Taylor series of the exponential function as
(14.213)
(14.214)
and use the Riemann formula
(14.99)
to obtain
where
y*
is the incomplete gamma function.
14.5.6
Differintegral
of
In(
Z)
For
all
values
of
q
the differintegral of ln(z)
is
given
as

(14.216)
where
y
is
the Euler constant, the value of which is
0.5772157,
and the
$(x)
function is defined
as
(14.217)
MATHEMATICAL TECHNIQUES WITH DIFFERINTEGRALS
413
14.5.7 Some Semiderivatives and Semi-integrals
We conclude this section with
a
table
of
the frequently used semiderivatives
and semi-integrals of some functions:
I
f
I
d4
f
/[dx]
3
I
d-i
f/[&]-$

I
14.6 MATHEMATICAL TECHNIQUES WITH DIFFERINTEGRALS
14.6.1 Laplace Transform of Differintegrals
The Laplace transform of
a
differintegral is defined
as
(14.218)
When
q
takes integer values, the Laplace transforms
of
derivatives and inte-
grals are given
as
(14.219)
dxq
k=O
E
-
=SqE{f},q=0,-1,-2
,
{
Z}
We can unify these equations
as
(14.220)
n-
1
(0),

n
=
O,fl,&2,f3,
_
k=O
(14.221)
414
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
In this equation we can replace the upper limit in the sum by any number
greater than
n-1.
We
are
now going
to
show that this expression is generalized
for
all
q
values
as
(14.222)
where
n
is an integer satisfying the inequality
n
-
1
<
q

5
n
.
as
We first consider the
q
<
0
case.
We write the Riemann-Liouville definition
and use the convolution theorem
where we take
fl(z)
=
x-~-'
and
f~(x)
=
f(x)
to write
(14.223)
(14.224)
(14.225)
=
sQL{f}.
For the
q
<
0
values the

sum
in Equation
(14,222)
is empty. Thus we
see
that
the expression in Quation
(14.222)
is valid for all
q
<
0
values.
For the
q
>
0
case we write the condition
[Eq.
(14.40)]
that the Griinwald
and Riemann-Liouville definitions satisfy as
(14.226)
where
n
is positive integer, and choose
n
as
n-l<q<n. (14.227)
We now take the Laplace transform

of
Equation
(14.226)
to find
dq-nf
n-l
=SnL{-}-~sk~
[z]
(0),
q-n<0.
(14.228)
k=O
dxqpn
MATHEMATICAL TECHNlQUES WITH DIFFERINTEGRALS
415
Since
q
-
n
<
0,
from Equations (14.223-225) the first term on the right-hand
side becomes
sqE{f}.
When
n
-
1
-
k

takes integer values, the term,
dn-
1-
k
-
&n-I-k
[-I
dxq-n
(O),
under the summation sign, with the
q
-
n
<
0
condition and the composition
formula [Eq. (14.226)], can
be
written
as
dq-1-k
dxq-
1-k
'
(O),
which leads
us
to
n-
1

dq-1-k
k=O
'(O),
0
<
q
#
1,2,3

.
(14.229)
dxq-1-k
We could satisfy this equation for the integer values
of
q
by taking the condi-
tion
n
-
1
<
q
5
n
instead of Equation (14.227).
Example
14.4.
Heat transfer equation:
We consider the heat transfer
equation for

a
semi-infinite slab:
(1 4.230)
K
is
the heat transfer coefficient, which depends on conductivity, den-
sity, and the specific heat
of
the slab. We take
T(z,t)
as
the difference
of the local temperature from the ambient temperature;
t
is the time,
and
x
is
the distance from the surface of interest.
As
the boundary
conditions
we
take
T(x,O)
=
0
(14.231)
and
T(m,t)

=
0.
(14.232)
Taking
the
Laplace transform
of
Equation (14.230) with respect to
t
we
get
aqx, s)
sF(x,
s)
-
T(x,
0)
=
K
ax2
'
(14.233)
(14.234)
416
FRACTIONAL DERIVATIVES AND INTEGRALS: "DIFFERINTEGRALS"
-
a2F(x,
s)
sT(x,
s)

=
K
ax2
.
(14.235)
Using the boundary condition [Eq.
(14.232)]
we can immediately write
the solution, which
is
finite
for
all
x
as
F(x,
s)
=
F(s)e-"rn,
(14.236)
In this solution
F(s)
is
the Laplace transform
of
the boundary condition
T(0,t)
:
F(s)
=

'
{T(O,t))
?
(14.237)
which remains unspecified.
In
most of the engineering applications we
are interested in the heat
flux,
which is given
as
(14.238)
where
k
is the conductivity. In particular, the surface
flux
given by
(14.239)
For
the surface
flux
we differentiate Equation
(14.236)
with respect to
XaS
and eliminate
F(
s)
by using Equation
(14.236)

to get
&FT(x,
s).
(14.240)
(14.241)
We now use Equation
(14.229)
and choose
n
=
1:
Using the other boundary condition [Eq.
(14.231)]
the second term on
the right-hand side is zero; thus we write
(14.243)
=
s'/2F(z,
s).
Substituting Equation
(14.241)
into this equation and taking the inverse
Laplace transform we get
(14.244)
MATHEMATICAL TECHNIQUES
WlTH
DIFFERINTEGRALS
417
Using this in the surface heat flux expression we finally obtain
J(0,

t)
=
-k
-
"2
t,
(14.245)
(14.246)
The importance of this result is that the surface heat flux
is
given in
terms
of
the surface temperature distribution, that is
T(0,
t),
which is
experimentally easier to measure.
14.6.2
Extraordinary Differential Equations
An equation composed of the differintegrals
of
an unknown function
is
called
an extraordinary differential equation. Naturally, solutions
of
such equations
involve some constants and integrals.
A

simple example of such an equation
can
be
given
as
(14.247)
Here
Q
is any number, F(x) is
a
given function, and f(x) is the unknown
function.
For
simplicity we have taken the lower limit
a
as
zero. We would
like to write the solution of this equation simply
as
(14.248)
dQ
dx-Q dxQ
and
-
are not the inverses
However, we have seen that the operators
-
of
each other, unless condition
d-Q

(14.249)
is satisfied. It
is
for this reason that extraordinary differential equations
are
in general much more difficult to solve.
A
commonly encountered equation in science
is
(14.250)
For
n
=
1
the solution
is
given as an exponential function
x(t)
=
zoexp(-at).
(14.251)
For
n
#
1
solutions are given with
a
power dependence
as