Internat. J. Mah. & Math. Sci
Vol. 5 No. 4 (1982) 785-791
785
A DIRECT EXTENSION OF MELLER’S CALCULUS
E.L. KOH
University of Petroleum and Minerals
Dhahran, Saudl Arabia
(Received June 26, 1980)
ABSTRACT. This paper extends the operational calculus of Meller for the operator
-e d +i d
t
Be
t
d- to the case where e e (0, oo). The development is la
Mikuslnskl calculus and uses Meller’s convolution process with a fractional derlva-
tlve operator.
KEY WORDS AND PHRASES. Operational Calculus, Mikusinski Calculus, Bessel Operator,
Convolution Proems, Fractional Derivative.
1980 MATHEK4TICS SUBJECT CLASSIFICATION CODES.
1.
44A40, 44A35, 33A40, 26A33, 93C50.
INTRODUCT ION.
Familiarity with integral transforms of distributions is assumed
is accessible to readers familiar with references
[1-3].
This paper
The generalist reader
interested in this area may start with these references.
Meller [4], [5] constructed an operational calculus for the operator
B
e
t-e
t te+l d-d
quotients.
with -i < e < 1 by embedding it in a field of convolution
The convolution process was given by the formula:
f(t)*g(t)
P(l + e)P(l
(t
e) dt
i
x
)-ed
d x
0
dx
0
e (1
x) ef (x)g[(1
x) (E
n)]
(1.1)
0
This calculus reduces to Ditkin’s calculus
[6], [3] for
d
t
d
when e
0.
Recently, Koh [7], [8] and Conlan [9] extended Meller’s calculus to the case
e e (-i, o).
to
Meller’s.
A modified convolution process was used which yields results analogous
In the present work, we give a direct extension of Meller’s calculus
by treaing the operator
vat lye.
1
F(I
d
e) dt
I
1
(t
0
)-d
in (i.i) as a fractional derl-
E.L. KOH
786
>
Specifically, we let n be the least integer greater than
times differentiable function
d
and i
For any n-
f(t), the th-order derivative of f(t) is:
Df(t)
where D is
O.
D
n
I
n-
(1.2)
f(t)
> 0 given in
is the Riemann-Liouville integral of order
Ross [i0] by
lf (t)
It is easy to see that I
F()
(1.3)
f()d.
6)
(t
J0
satisfies the semigroup property
ii= i +
but D
Thus
does not.
DaD
(1.4)
in (2.1) below cannot be written D
+I.
THE CONVOLUTION QUOTIENTS.
2.
_>
Let
ly differentiable functions on [0,oo).
in C
Let C
0 be a fixed real number.
denote the linear space of infinite-
For every pair of functions (t) and (t)
define their convolution by
(t)*(t)
1
DtD
F( + i)
It I
0
I
N(I
(xD)[(l
x)
From this definition, the following properties are clear:
x C
convolution, (ii) convolution is bilinear on C
(1) C
(x)
(v,O
Let x
-
Convolution is commutative.
i
i for all t g
(t)*(t)
< i and to Ditkin’s
Not so immediate are the following properties.
PROPOSITION I.
PROOF.
is closed under
It also follows immediately
that equation (2.1) specilizes to Meller’s convolution for
0.
1
F(a + i)
1
F( + i)
V
and N
(0, )
t
t
in (2.1) and noting that the Jacobian
we have
[lito(t t)a( vq
i
DDtD t
J v(l )(v)[(l-
DeDtD
[(i
v
)(t
0
v
t)][
(t)]dvd
)(t- v)]ddv
00
(t)*(t).
PROPOSITION 2.
PROF.
(2.1)
(iii) convolution is distri-
butive with respect to the usual addition of functions.
for
N)]dxdN
x)(t
0
l*(t)
q.e.d.
For every complex nber %, and any #(t) e C
F(e + i)
DDtD
F(a + i)
Dta+l
(i0
x)l(xN)dxdN
0
i
0
(i
x)a(xt)dx
%*(t)
%(t).
EXTENSION OF
r( + 1)
DaD
r(+ l)
Dnln-
MELLER’ S
CALCULUS
787
u)(u)du
t
0
t
u)
(t
(u)du
0
%Dnln-ala(t)
%(t).
The last step follows from (1.4) and (1.2). q.e.d.
In view of Proposition 2, there
is no distinction between constants and constant functions in our calculus.
PROPOSITION 3.
PROOF.
Convolution is associative.
A direct calculation shows that, for nonnegative integers q and r,
q!r!r(q + + l)r(r + + i).
(q + r)!r( + l)r(q + r + + i)
tq,t r
t
q+r
(2.2)
Hence on using (2.2) again,
p!q!r! r(p+a+l)!r(q+a+l)r(r+a+l}
(p+q+r) F (0i) F (p+q+r++l) F (+i)
tP,(tq,tr
(t
p
,
,
t q)
t
tp+q+r
r
(2.3)
Due to the bilinearity of our convolution, equation (2.3) still holds for polynomOur proposition follows from Weierstrass’s Approximation Theorem and the
ials.
fact [9] that the space of C
PROPOSITION 4.
and
(t) * (t)
(t
)
As t
-
Cn
O.
(t)
0.
C_l
tn-i
+
tn-2
C2
(n
2)’.
Now, by an argument leading
to
(2.3),
(n
i)!
some i, then (t) and (t) have to be polynomials.
n)]dxdN
+
+
Cn
we see that, if C
0
0
x)aq(xrl)[ (I
x)(t
i
# 0 for
Hence, the right side of (2.4)
A similar argument, together with Titchmarsh’s Theorem
it ll rlc(1
(2.4)
But if they are polynomials,
the left side of (2.4) will be of degree at least n.
has to be zero.
x)(
(xr])[(l
x)
0
0
C1
O,
0 or
0 implies that
-F
0
q.e.d.
has no zero divisors, i.e. if (t) and (t) belong to C
C
0, then either (t)
(t) * (t)
PROOF.
functions with compact support is dense in C
ri)]dxdr]
O.
2], yields
(2.5)
E.L. KOH
788
z and
To complete the proof, let x
Ivl
0
-(’r
v in (2.5).
t
Oo
(y.)q[(’r- y)(v- .)]dyd-
y)
We then have
0
z(yz)
By a theorem of kuslnsi and Ryll-Nardzewski [ii], it follows that
y(yz)
O.
0 or (t)
Thus, (t)
0.
The above properties establish C
q.e.d.
as an integral don der the operations
By virtue of Proposition 2, the
of addition and convolution as multplcaton.
s
multpllcatve dentlty for C
0 or
nto the
We may now extend C
the nber i.
field F of convolution quotients consisting of equivalence classes of ordered pairs
n
(,) of elements
C
# 0.
with
The equivalence relation is
As usual, convolution quotients are called operators
Operators of the form
(t)
the canonical maps
3.
.(t)
-+
by
and are denoted by
constitute a subring of F isomorphic to C
through
(t)
AN OPERATIONAL CALCULUS.
belongs to F.
We now show that the operator B
inverse to
B
BA
i.e.
A
na(n)dnd;
,
0
0
for
g
C
PROPOSITION 5
.
If we restrict the domain of
ABa
is also a left inverse; i.e.,
PROOF.
First, note that a right
is given by
A
then
2
gven
For any (t)
g
t
C
+i
* (t)
We shall assume that a is not an integer.
-
B
to
{
I(0)
g
0},
A(t)
Otherwise, the proof is
more straightforward, obviating the use of fractional integrals.
t
e + 1
* qb(t)
i
F(o + i)
1
F(e + 2)
F(a + 2)
F( + I)
DDtD
DeDtD
l ll n( (i- x) c (x)
t
)u )u
Dc{
n
a
2
(rl-$)
On
D
n)
(t
0
(n-
)a+ l($)ddn
0
()ddn +
0
fto- frl
i
o
(I- x)(t- )
dxdn
o + 1
(t
)
()d}
0
(r- )
ot ()ddn
(3.1)
EXTENSION OF MELLER’ S CALCULUS
In()
Let ()
t
+
i
least integer greater than a.
where n
* (t)
D
it i
F(
0
n
It
_(t
(n
0
+ i’) ()dd
n
lu i .-
)n-a-i
u
)
1
a)
F(a
0
0
n
a-n
+ i) ()ddNdu
I I rl-l(rl
n
t
t
D
rl)n-tdrld.
)c-n(t
()
F (n-e+l) F (a-n+l)
Then (3.1) becomes
-n
)
i
0
D
789
0
{()-n-l}F(-n+l)F(n- ).
The inner integral reduces, via the Beta function, to
Thus,
t
+
i
* (t)
n
1}d
)(J){(--
O.
0
fl
Dn
-n
n- 0 t(wt)(w
l)dw
-n
0
ft
O
(wn--i (wnto(wt) + nwn-llO(wt))dw
1
n
t)
c [(
()n]($)d
-It
0
(
-e
)
[(
(x()
rl
()hi(
)
[(
1]d
-c-I
drld
$ ()ddr
This result implies that operators of the form
A(t).
(t)_
t
with locally integrable functions f(t) such that
f(t) e
Llo c[0, oo)
(u)dud
0
0
0
..O(t).t
.
iff
O(t)
q.e.d.
with #(0)
Af(t) <
0 may be identified
for every t > O.
(e + 1) A f < oo,
t*f(t)
Vt > 0.
The next result follows from Proposition 5 and Equation (2.2) by induction.
PROPOSITION 6.
Let k be a positive integer.
k
Let V be
n!t
Ak(t)
(k + n) lk! * 0(t)
+ i and Vk the
the operator
t
where
Then, for any #(t) e C
Ak(t)
A(A(-.- (A))).
k-times
k-times application of V.
Indeed,
790
E.L. KOH
PROPOSITION 7.
For any 2k times differentiable function (t),
k
(3.2)
j--1
PROOF.
V(t)
ABa(t)
(t)
+ (0)V
Ba(t)
true for k
+ (0)
+ (0).
Ba(t)
+ I
a
and (3.2) is proved for k
i.
Thus
Suppose now that (3.2) is
Then for any 2m times differentiable function
m- i.
(t),
m-i
m-l
Vine(t)
V(Bo
Z Bm-l-Jc (1)(t) lt_+
(t) +
Vj)
j=l
m-i
B Bm-1 *(t) + Bm-1 *(t)
a a
It+O+
V +
m
B
Z Bm-l-J*(t)
j=l
.^+
"t-
Vj+l
m
]--1
The proposition follows by induction.
A number of operational formulas such as those in Theorems 5 and 6 of [7]
may
be generated by using (3.2).
The proofs are similar, mutatis mutandis.
A generali-
zation of Theorem 5 of [7] is obtained by parametric differentiation.
P( + I)
V
PROPOSITION 8.
(V
a)
m+l
m!
tin(at)- --i la+m (2k/)
&+m
where
l(x)
J(x)
and
RHARKS.
i.
V
F(a + I)
(V + a) m+l
m!
t
m(at)--T" J&+m (2
are Bessel functions of order
All the results of Meller are extendible to the case
(0,)
via the metho given in this paper.
2.
The operational calculus may be applied to certain time-varying systems
and to Kratzel’s problem as done in [8].
3.
In [12], a convolution for the operator A
t
-n-1 d
d-
t
n+l d
d--[
where n is
a natural number, is given which is associative, commutative, and distributive with
respect to addition.
However, the ring under convolution as multiplication contains
zero divisors.
ACKNOWLEDGEICENT.
The author is on leave of absence from the University of Regina,
Regina, Canada.
The research is partly supported by the National Research Council
of Canada ’under a grant numbered A-7184.
EXTENSION OF
MELLER’ S
CALCULUS
79i
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1.
ZEMANIAN, A.H. .istrlbution Theory and Transform Analysis, McGraw-Hill,
New York, 1965.
2.
MIKUSINSKI, J.
3.
DITKIN, V.A. and PRUDNIKOV, A.P. Integral Transforms and Operational Calcul.us.,
Pergamon Press, New York, 1965.
-d
+I d
t
an operational calculus for the operator
MELLER, N.A.
dt
Vichishitelnaya Matematika 6 (1960), 161-168.
4.
Operational Calculus, Pergamon Press, Oxford, 1959.
n
B--t
Some applications of operational calculus to problems in analysis, Zh. Vichisl. Mat. i. lt. Fiz. 3(i) (1963), 71-89.
5.
MELLER, N.A.
6.
DITKIN, V.A. Operational calculus theory, Dokl. Akad. Nauk. SSSR. ii__6
(1957), 15-17.
7.
KOH, E.L.
8.
KOH, E.L.
9.
CONLAN, J. and KOH, E.L. On the Meijer transformation, Int. J. Math. and
Math. Sc. i (1978), 145-159.
i0.
ROSS, B. (ed.) Fractional Calculus and its Applications, Springer-Verlag,
Berlin, 1975.
ii.
MIKUSINSKI, J. and RYLL-NARDZEWSKI, C.
A Mikusinski calculus for the Bessel operator
Springer-Verlag Lect. Notes #564 (1976), 291-300.
B,
Proc. Diff. Eq.,
Application of an operational calculus to certain time-varying
systems, Int. J. Systems Sc. 9 (1978), 595-601.
Un theorm sur le product de composition des fonctions de plusieurs variables, Studla Math. 13(i) (1953),
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12.
CHOLEWINSKI, F.M.
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