International Journal of Trend in Scientific
Research and Development (IJTSRD)
International Open Access Journal
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 5
Fractional Quadruple Laplace Transform and its Properties
1
Monisha. G1, Savitha. S2
Research Scholar, 2Faculty of Mathematics
Department of Mathematics, Vivekanandha College of Arts and S
Sciences
ciences for Women [Autonomous],
Trichengode, Namakkal
Namakkal, Tamil Nadu, India
ABSTRACT
In this paper, we introduce definition for fractional
quadruple Laplace transform of order α,0 < α ≤ 1, for
fractional differentiable functions. Some main
properties and inversion theorem of fractional
quadruple Laplace transform are established. Further,
the connection between fractional quadruple Laplace
transform andd fractional Sumudu transform are
presented.
KEYWORD: quadruple Laplace transform, Sumudu
transform, fractional Difference.
INTRODUCTION
There are different integral transforms in mathematics
which are used in astronomy, physics and also in
engineering. The integral transforms were vastly
applied to obtain the solution of differential equations;
therefore there are different kinds of integral
transforms like Mellin, Laplace, and Fourier and so
on. Partial differential equations are considered one of
the most
ost significant topics in mathematics and others.
There are no general methods for solve these
equations. However, integral transform method is one
of the most familiar methods in order to get the
solution of partial differential equations [1, 2]. In [3,
9] quadruple Laplace transform and Sumudu
transforms were used to solve wave and Poisson
equations. Moreover the relation between them and
their applications to differential equations have bee
been
determined and studied by [5, 6]. In this study we
focus on quadruple
druple integral determined and studied by
[5, 6]. In this study we focus on quadruple integral
transforms. First of all, we start to recall the definition
of quadruple Laplace transform as follows
follows.
Definition: Let be a continuous function of four
variables; then the quadruple Laplace transform of
, , , is defined by
(, ), *
,
, , , (1)
∞ ∞ ∞ ∞
&
, , , 9999
Where , , , 0
variables, and
, , , and
, (, ), *are Laplace
%!"∞
!"∞
:!"∞
1
1 $
1 $
1 $
#
#
#
2
2
2
2
"∞
'!"∞
%"∞
:"∞
& , (, ), * $9;$9;$9;9
'"∞
is the inverse Laplace transform.
Fractional Derivative via Fractional Difference
Definition: let +, be a continuous function, but not
necessarily differentiable function, then the forward
operator -(h)
(h) is defined as follows -h +, +, . / , Where / 0 denote a constant
discretization span.
Moreover, the fractional difference of +, is known
as
∆ +, - 1 / +,
∞
6
2 11 3 5 8 +,
+ . 6 1 7 /
7
34
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Where 0 < 6 < 1,
And the α-derivative of+(,) is known as
△ +(,)
+( ) (,) = lim
A↓
ℎ
See the details in [9, 10].
Fractional quadruple Sumudu Transform
Recently, in [13] the author defined quadruple
sumudu transform of the function depended on two
variables. Analogously, fractional quadruple sumudu
transform was defined and some properties were
given as the following
Modified Fractional Riemann-Liouville Derivative
The author in [10] proposed an alternative definition
of the Riemann-Liouville derivative
Definition: let +(,) be a continuous function, but not
necessarily differentiable function, then
Let us presume than +(,) = D , where D is a
constant, thus α-derivative of the function +(,)
, 6 ≤ 0, $
DΓH (1 − 6)t isEF D = G
0,
K,ℎ)*.
On the other hand, when +(,) ≠ D hence
+(,) = +(0) + N+(,) − +(0)O,
and fractional derivative of the function +(,) will
become known as
+( ) (,) = EF +(0) + EF N+(,) − +(0)O,
at any negative α, (α < 0) one has
EF N+(,) − +(0)O
F
1
=
(, − P)
Γ(−α)
H
+(P)9P, 6 < 0,
While for positive α, we will put
EF N+(,) − +(0)O = EF +(,) = EF N+( H) O.
When 7 < 6 < 7 + 1, we place
+( ) (,) = (+( 3) (,))(3) ,
7 ≤ 6 < 7 + 1, 7 ≥ 1.
Integral with Respect to (RS)T
The next lemma show the solution of fractional
differential equation
9 = +(,)(9,) , , ≥ 0, (0) = 0
By integration with respect to (9,)
(2)
Lemma: If +(,) is a continuous function, so the
solution of (2) is defined as the following
H
(,) = +(P)(9P) , (0) = 0
H
= 6 (, − P)
+(P)9P, 0 < 6 < 1
∞
∞
∞
∞
= [ (−( + + + ,) )(, (, ), *,)(9) (9) (9) (9,)
Where, (, ), * ∈ ℂ, , , , , > 0
and
3
[ () = 2
34 Γ(αm + 1)
is the Mittag-Leffler function.
∞
Some Properties of Fractional Quadruple Sumudu
Transform
We recall some properties of Fractional Quadruple
Sumudu Transform
V W X (^, _, `, 9,)Y = Z W (^, _(, `), 9*),
V W X ( − ^, − _, − `, , − 9)Y
= [ (−(^ + _ + ` + 9) )Z W (^, _(, `), 9*),
V W XaF (^, _, `, 9,)Y
Z W (, (, ), *) − Γ(1 + 6)(0, , , )
(4)
=
b
where aF is denoted to fractional partial derivative of
order α (see [10]).
Main Results
The main results in this work are present in the
following sections
Quadruple Laplace Transform of Fractional order
Definition 6: If (, , , ,) is a function where
, , , , > 0, then Quadruple Laplace Transform of
Fractional order of (, , , ,) is defined as
W X(,
, , ,)Y = W (, (, ), *)
(5)
∞
∞
∞
∞
= [ (−( + ( + )
H
Definition: [14] The fractional quadruple sumudu
transform of function (, ,) is known as
V W X (, , , ,)Y = Z W (, (, ), *),
(3)
For more results and varies views on fractional
calculus, see for example [15, 16, 17, 18, 19, 20, 21]
+*,) )(, , , ,)(9) (9) (9) (9,)
Where , (, ), * ∈ ℂ and [ () is the Mittag-Leffler
function.
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
Corollary 10.1 By using the Mittag-Leffler property
then we can rewrite the formula (5) as the following:
W X(,
, , ,)Y = W (, (, ), *)
= [ (−() )[ (−( )[ (−()) )
∞
∞
∞
∞
[ (−(*,) )(, , , ,)(9) (9) (9) (9,)
(6)
Remark 8: In particular case, fractional quadruple
Laplace transform (5) turns to quadruple Laplace
transform (1) when α = 1.
Some properties of fractional quadruple Laplace
transform
In this section, various properties of fractional
quadruple Laplace transform are discussed and proved
such as linearity property, change of scale property
and so on.
1. Linearity property
Let H (, , , ,) and d (, , , ,) be functions of the
variables and ,, then
W X^
H H (, , , ,) + ^d d (, , , ,)Y
=^H W XH (, , , ,)Y + ^d W Xd (, , , ,)Y(7)
Where ^H and ^d are constants.
Proof: We can simply get the proof by applying the
definition.
2. Changing of scale property
W X(,
If
, , ,)Y = W (, (, ), *)
W X(^,
_, `, 9,)Y
1
( ) *
=
W 5 $ , , , $ 8
^ _ ` 9
^ _ ` 9
∞
∞
hence
∞
= [ (−( + ( + )
+ *,) )(^, _, `, 9,) (9) (9) (9) (9,) (8)
We set f = ^ , g = _, h = ` and 7 = 9, in the
above
equality (8),
therefore
we
obtain
W X(^,
_, `, 9,)Y
∞ ∞ ∞ ∞
1
f (g )h
[ i− i +
+
^ _ ` 9 ^
_
`
*7
+ $ 8 j$ (f, g, h, 7)(9) (9) (9) (9,)
9
=
1
( ) *
W 5 $ , , , $ 8
^ _ ` 9
^ _ ` 9
3. Shifting property
W X(,
Let
, , ,)Y = W (, (, ), *)
then
W
X[ (−( + ( + ) + *,) )(, , , ,)Y
= W ( + ^, ( + _, ) + `, * + 9)
Proof:
W
X[ (−(^ + _ + ` + 9,) )(, , , ,)}
∞
∞
∞
∞
= [ (−(^ + _ + `
+ 9,) )[ (−( + ( + )
+ *,) ) (, , , ,)(9) (9) (9) (9,)
By using the equality
X[ (k( + + + ,) )
=[ (k ) [ (k )[ (k )[ (k, )
Which implies that
W
X[ (−(^ + _ + ` + 9,) )(, , , ,)
= [ (−((^ + ) + (_ + () + (`
∞
∞
∞
∞
+ ))(9 + *),) ) (, , , ,)(9) (9) (9) (9,)
Hence
W
=
Whenever^ and _ are constants.
Proof:
X(^, _, `, 9,)Y
∞
=
W (
X[ (−( + ( + ) + *,) )(, , , ,)Y
+ ^, ( + _, ) + `, * + 9)
4. Multiplication by ,
W X(,
, , ,)Y = W (, (, ), *)
∞
∞
∞
∞
= [ (−() )[ (−(() )[ (−()) )
[ (−(*,) )(, , , ,)(9) (9) (9) (9,)
then
ad
X , (, , , ,)Y =
a a( a) a*
W X(,
, , ,)Y
Proof:
∞
∞
∞
W X
∞
, (, , , ,)Y
= [ (−() )[ (−(() )[ (−()) )
[ (−(*,) )(, , , ,)(9) (9) (9) (9,)
By using the fact E& ([ (−* )) =
[ (−* ), then
W X
-
, (, , , ,)Y
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
a
a
Proof: We obtain the definition of fractional
[ (−() )
[ (−(() )
quadruple Laplace transform and fractional quadruple
a(
a
convolution above, then we obtain
a
a
W X(
(9,)
[ (−())
[ (−(*,) )(, , , ,)(9) (9) (9)
∗∗∗∗ +)(, , , ,)Y
a)
a*
∞
∞ ∞ ∞
∞ ∞ ∞ ∞
W
a
= [ (−() )[ (−(()=) [ (−() )[ (−(() )[ (−()) )
a a( a) a*
[
(−(*,)
)(
[ (−()) )[ (−(*,) )(, , , ,)(9) (9) (9) (9,) ∞ ∞ ∞ ∞ ∗∗∗∗ +)(9) (9) (9) (9,)
ad
= [ (−() )[ (−(() )[ (−()) )(9,)
W X(,
=
, , ,)Y
a a( a) a*
F
[ (−(*,) ) s ( − P, − p, − q, ,
Remark: All results above are suitable for quadruple
Laplace transform when α = 1.
− r)+(P, p, q, r)(9P) (9p) (9q) (9r) t (9)
∞
∞
∞
= ∞
Theorem: If the fractional quadruple Laplace
transform of the function (, , , ,) is
W X(,
, , ,)Y = W (, (, ), *),
and
fractional
quadruple sumudu transform of the function
(, , , ,) is V W X (, , , ,)Y = Z W (, (, ), *),then
1 1 1 1
1
Z W (, (, ), *) =
W i $ , , , $ j
( ) *
( ) *
Proof:
Z W (, (, ), *)
∞
∞
∞
∞
= [ (− )[ (− )[ (− )[ (−, )
(, (, ), *,)(9) (9) (9) (9,)
(9) (9) (9,)
let f = − P, g = − p, h = − q, 7 = , − r
taking the limit from 0 to ∞, it gives
∞
∞
∞
and
∞
= [ (− (f + P) )
[ (−( (g + p) )[ (−) (h + q) )[ (−* (7
F
+ r) ) (f, g, h, 7)+(P, p, q, r)(9P)
(9p) (9q) (9r) (9f) (9g) (9h) (97)
∞
∞
∞
∞
= [ (− f )[ (−( g )[ (−) h )
By using change of variables f → , g → (, h → )
[
(−*
7
)(f,
g, h, 7)(9f) (9g) (9h) (97)
7→9
and
F
∞ ∞ ∞ ∞
1
f
g
h [ (− P )[ (−( p )[ (−) q )
i
=
[ m− i j n [ m− i j n [ m−
j n
) ( ) * (
[ (−* r )+(P, p, q, r)(9P) (9p) (9q) (9r)
7
W X(,
[ i− 5 8 j (f, g, h, 7)(9f) (9g) (9h) (97)
, , ,)Y W X(, , , ,)Y
*
1
1 1 1 1
Z W (, (, ), *) =
W i , , , j
Inversion formula of Quadruple Fractional
( ) *
( ) *
Laplace’s transform
The Convolution Theorem of the Fractional Firstly, we will set up definition of fractional delta
function of two variables as follows
Double Laplace Transform
Theorem: The double convolution of order α of
functions(, , , ,) and +(, , , ,) can be defined
as the expression
((, , , ,) ∗∗∗∗ +(, , , ,))
F
= ( − P, − p, − q, ,
− r)+(P, p, q, r) (9P) (9p) (9q) (9r) ,
therefore
one
has
the
equality
W X(
∗∗∗∗ +)(, , , ,)Y
= W X(, , , ,)Y W X(, , , ,)Y
Definition: Two variables delta function r ( −
^, − _, − `, , − 9) of fractional order α, 0 < α ≤1,
can be defined as next formula
+(, , , ,) r ( − ^, − _, − `, ,
u u u
u
− 9)(9) (9) (9) (9,)
= 6 W +(^, _, `, 9)(9)
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
[ ( (ℎ − ) )[ ( (b − () )
In special case, we have
+(, , , ,) r (, , , ,)(9) (9) (9) (9,)
ℝ ℝ ℝ ℝ
[ ( (w − )) [ (, ( − *)
(9) (9() (9)) (9*)
u u u
u
W
= 6 +(0,0,0,0)
Example: we can obtain fractional quadruple Laplace
transform of function r ( − ^, − _, − `, , − 9)
as follows
W Xr (
− ^, − _, − `, , − 9)Y
∞
∞
∞
∞
= [ (−( + ( + ) + *,) )
r ( − ^, − _, − `, , − 9)(9) (9)
(9) (9,)
= 6 W [ (−(^ + ( + ) + *,) )(10)
In particular, we have W Xr (, , , ,)Y = 6 d
Relationship between Two Variables Delta
Function of Order α and Mittag-Leffler Function
The relationship between [ ( + + + ,) and
r (, , , ,) is clarified by the following theorem
= (9) (9) (9) (9,)
ℝ ℝ ℝ ℝ
6W
[ ((−f) )[ ((−g) )
(v )W
ℝ ℝ ℝ ℝ
[ ((−h) )[ ((−,7) )(9f) (9g) (9h) (97)
= r (, , , ,)(9) (9) (9) (9,)
ℝ ℝ ℝ ℝ
d
=6
Note that one has as well
6W
[ ((−ℎ) [ ((−b) [ ((−w)
(v )W
ℝ ℝ ℝ ℝ
[ ((−,) (9ℎ) (9b) (9w) (9) = r (, , , ,)
Inversion Theorem of Quadruple Fractional
Theorem:
The
following
formula
holds Laplace Transform
6W
Theorem: Here we recall the fractional quadruple
[ ((−ℎ) [ ((−b) [ ((−w)
W
Laplace
transform
(5)
for
convenience
(v )
W X(,
W (,
ℝ ℝ ℝ ℝ
, , ,)Y = (, ), *)(12)
[ ((−,) (9ℎ) (9b) (9w) (9)
∞ ∞ ∞ ∞
= r (, , , ,)(11)
= [ (−( + ( + )
Wherev satisfy the equivalence [ (((v ) ) =1,
and it is called period ofthe Mittag-Leffler function.
Proof: We test that (11) agreement with
6 d = [ ((ℎ) [ ((b) [ ((w)
ℝ ℝ ℝ ℝ
[ ((,) (9) (9) (9) (9,)
We replace r (, , , ,)in above equality by (11) to
get
6W
6 d = (9) (9) (9) (9,)
(v )W
ℝ ℝ ℝ ℝ
[ ((ℎ) )[ ((b) )[ ((w) )
ℝ ℝ ℝ ℝ
[ ((,) )[ ((−) [ ((−() [ ((−))
[ ((−*,) (9) (9() (9)) (9*)
= (9) (9) (9) (9,)
ℝ ℝ ℝ ℝ
6
(v )W
W
+ *,) )(, , , ,)9 9 9 9,
And its inverse formula define as
(, , , ,)
∞ ∞ ∞ ∞
1
=
[ (( + ( + )
(v ) W + *,) ) W (, (, ), *)(9) (9() ( 9)) (9*)
(13)
Proof:
Substituting (12) into (13) and using the formula (11),
(9) respectively, we obtain in turn
(, , , ,) =
∞
1
(v
)
!"∞ !"∞ !"∞ !"∞
W
[ ()
"∞ "∞ "∞ "∞
[ (() [ ()) [ (*,) (9) (9() (9)) (9*)
∞
∞
∞
[ (−(f + (g + )h + *7)
(f, g, h, 7)(9f) (9g) (9h) (97)
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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
1
=
(y, q, z, p) (9y) (9q)
(v )W
References
1. D. G. Duffy, Transform methods for solving
partial differential equations (CRC press, 2004)
(9z) (9p)
2. T. A. Estrin and T. J. Higgins, Journal of the
Franklin Institute 252, 153-167(1951)
∞ ∞ ∞ ∞
!"∞ !"∞ !"∞ !"∞
[ ( ( − f) )
"∞ "∞ "∞ "∞
[ (( ( − g) )[ () ( − h) )[ (* (* − 7) )
(9) (9() (9)) (9*)
1
(v )W
=
(y, q, z, p)
(v )W
6W
∞ ∞ ∞ ∞
r (f − , g − , h − , 7 − *)(9y)
(9q) (9z) (9p)
1
= W (y, q, z, p) r (f
6
∞ ∞ ∞ ∞
− , g − ,
h − , 7 − *)(9y) (9q) (9z) (9p)
= (, , , ,)
Conclusion
In this present work, fractional quadruple Laplace
transform and its inverse are defined, and several
properties of fractional quadruple transform have been
discussed which are consistent with quadruple
Laplace transform when α = 1. More over convolution
theorem is presented.
3. A. Kilicman and H. Gadain, Journal of Franklin
Institute 347,848-862(2010)
4. A. Kilicman and H. Eltayeb, Applied
Mathematical Sciences 4, 109-118(2010)
5. H. Eltayeband A. Kilicman, Malaysian Journal of
mathematical Sciences 4, 17-30(2010)
6. S. G. Samko, A. A. Kilbas, and O. I. Marichev,
Fractional integrals and derivatives, fractional
differential equations, to methods of their solution
and some of their applications, Vol. 198
(Academic press, 1998)
7. R. Hilfer, Applications of fractional calculus in
physics (World Scientific, 2000)
8. K. Oldhman and J. Spanier, The fractional
calculus: Theory and application of differentiation
and integration to arbitrary order198 (Academic
press, New York, 1974)
9. Hamood Ur Rehman, Muzammal Iftikhar, Shoaib
Saleem, Muhammad Younis and Abdul Mueed, A
Computational Quadruple Laplace Transform for
the Solution of Partial Differential Equations,
3372-3382.
10. T. J. Osler, SIAM Journal on Mathematical
Analysis 2, 37-48(1971).
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