New integral "Kuffi-Abbas-Jawad" KAJ transform and its application on ordinary differential equations

Abstract

In this work, a new integral transform called the "Kuffi-Abbas-Jawad" KAJ integral transform has been proposed and applied to solve ordinary differential equations. The proposed KAJ integral transform is a modification of the well-known (Sadiq-Emad-Eman) SEE integral transform. It is a novel method that could be used to solve ordinary differential equations. The KAJ integral transform has been tested and proven for its excellent ability to solve homogeneous and non-homogeneous higher-order ordinary differential equations.

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Journal of Interdisciplinary Mathematics
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/tjim20
New integral “Kuffi-Abbas-Jawad” KAJ transform
and its application on ordinary differential
equations
Elaf Sabah Abbas, Emad A. Kuffi & Alyaa Abedalrhman Jawad
To cite this article: Elaf Sabah Abbas, Emad A. Kuffi & Alyaa Abedalrhman Jawad (2022): New
integral “Kuffi-Abbas-Jawad” KAJ transform and its application on ordinary differential equations,
Journal of Interdisciplinary Mathematics, DOI: 10.1080/09720502.2022.2046339
To link to this article: https://doi.org/10.1080/09720502.2022.2046339
Published online: 04 Jul 2022.
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New integral “Kuffi-Abbas-Jawad” KAJ transform and its application
on ordinary differential equations
Elaf Sabah Abbas *
Department of Communication Engineering
Al-Mansour University College
Baghdad
Iraq
Emad A. Kuffi †
Department of Material Engineering
College of Engineering
Al-Qadisiyah University
Iraq
Alyaa Abedalrhman Jawad §
Department of Mathematics
College of Science
The Iraqia University
Baghdad
Iraq
Abstract
In this work, a new integral transform called the “Kuffi-Abbas-Jawad” KAJ integral
transform has been proposed and applied to solve ordinary differential equations. The
proposed KAJ integral transform is a modification of the well-known (Sadiq-Emad-Eman)
SEE integral transform. It is a novel method that could be used to solve ordinary differential
equations. The KAJ integral transform has been tested and proven for its excellent ability
to solve homogeneous and non-homogeneous higher-order ordinary differential equations.
Subject Classification: 44 (Integral transforms, Operational calculus).
Keywords: Differential equations, Modified SEE transform, Kuffi-Abbas-Jawad KAJ transform, SEE
integral transform.
* E-mail: elaf.abbas@muc.edu.iq (Corresponding Author)
† E-mail: emad.abbas@qu.edu.iq
§ E-mail: first_lady20062000@yahoo.com
Journal of Interdisciplinary Mathematics
ISSN: 0972-0502 (Print), ISSN: 2169-012X (Online)
DOI : 10.1080/09720502.2022.2046339

2 E. S. ABBAS, E. A. KUFFI AND A. A. JAWAD
1. Introduction
Differential equations (DEs) played an important role in many
scientific and engineering fields, for their importance in representing
the changes that could occur in scientific and engineering problems in
mathematical forms. DEs are mathematical statements that contain
derivatives in them and can be solved as equations containing constants
and variables dependent on one another, these equations represent
behavioral predictions for the original problems, [1,2]. Due to the
importance of differential equations, many methods have been proposed
and used to solve them, including mathematical transforms [3-7], and the
proposal for new methods to solve differential equations is not going to
stop any time soon.
In this paper a new integral transform called “Kuffi-Abbas-Jawad”
KAJ Integral Transform has been proposed, which is a modification on
the (Sadiq-Emad-Eman) SEE integral transform, that proposed in (2021)
[8,9], the function f that used in KAJ transform depends on two variables t
and v, instead of depending only on t variable, that modification made the
transform an efficient tool in signal processing and other applications that
contained decreasing in the original signal power.
2. SEE Integral Transform Basic Concepts
In the year (2021), Eman, Sadiq and Emad [8,9], defined “SEE Integral
Transform” of the function f (t) for
0t³
as:
{ ( )}S ft =
0
1( ) dt
vt
nt
vfte
¥-
=
ò
( ),Tv=
12
, , 0,n llÎ>
12
.l vl££
Where the operator S is called the SEE
transform operator.
2.1 SEE Integral Transform of Frequently Encountered Functions, [8,9]
•
1
{}
n
K
v
SK
+
=
, K is a constant.
•
1
!
{.
} ,
m
nm
m
St m
v
++
=Î

•
1
( 1)
}{ ,1
m
nm
m
St m
v
++
G+
= >-
.
•
1
()
{,}
at
n
v va
Se -
=
a is a constant.
•
22
{sin (at } .
)
)(
n
a
Svv a
=+

NEW INTEGRAL KAJ TRANSFORM 3
•
22
{cos(at } .
)
)(
n
v
Svv a
=+
•
22
{sinh (at)} .
)(
n
Svv a
a
=+
•
22
{cosh(at)} .
( ( ))
n
v
Svv a
=-
1.2 SEE Integral Transform of Derivatives, [8,9]
SEE integral transform for the
th
m
derivative is:
()
{ ( )}
m
Sf t =
( 1)(0)
m
n
f
v
-
--
( 2)(0)
1
m
n
f
v
-
--××× (0)
1
f
nm
v
-+
-
( ).
m
v Tv+
3. KAJ Transform Basic Concepts
KAJ transform (denoted by Sm), as a modification on the SEE
integral transform, is given by:
{ ( )} ( )
m
S ft Kv=
0
1dt,
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø
=ò
,nÎ
12
0 ,l vl<££
where l1 and l2 are either finite or infinite.
3.1 KAJ Integral Transform for Some Basic Functions
• Let
( ) 1,ft=
using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
0
11dt
t
ntv
vfe
¥-
=
æö
ç÷
èø =
ò
11
0
11
dt .
vt
nn
t
vv
e
¥-
++
==
ò
• Let
() ,ft t=
using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
11
0
11
dt .
t
nn
t
vv
te
¥-
++
==
ò
• Let
() ,
m
ft t= using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
0
1dt
mt
nt
t
v
ve
¥-
==
ò10
!1 dt , .
mt
n nm
tv
m
vte m
¥-
++
=
=Î
ò
• Let
() ,
at
ft e=
using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
0
1dt
at
t
v
nt
vee
¥-
==
ò0
11
1
()
1
dt .
at
tv
n nn
t
v
a
v v v va
v
e-+
¥
=
æö
ç÷
ç÷
ç÷ -
-
ç÷
èø
==
ò
• Let f (t) = sin (at), using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
0
1sin dt
t
nt
at
v
ve
¥-
=
æö
ç÷
èø =
ò12 2
22
1
()
.
nn
a
av
v v va
va -
æö
ç÷
ç÷ -
+
èø
=

4 E. S. ABBAS, E. A. KUFFI AND A. A. JAWAD
• Let f (t) = cos (at), using KAJ transform:
{ ( )}
m
S ft =
0
1dt
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø =
ò
0
1cos dt
t
nt
at
v
ve
¥-
=
æö
ç÷
èø =
ò2
22 2
22
11
()
.
nn
v
v v va
va -
æö
ç÷
ç÷ -
+
èø
=
• Let f (t) = sinh (at), using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø =
ò
0
1sinh dt
t
nt
at
v
ve
¥-
=
æö
ç÷
èø =
ò12 2
22
1
()
.
nn
a
av
v v va
va -
æö
ç÷
ç÷ -
+
èø
=
• Let f (t) = cosh (at), using KAJ transform:
{ ( )}
m
S ft
=0
1dt
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø =
ò
0
1cosh dt
t
nt
at
v
ve
¥-
=
æö
ç÷
èø =
ò2
22 2
22
11
()
.
nn
v
v v va
va -
æö
ç÷
ç÷ -
+
èø
=
3.2 KAJ Integral Transform of Derivatives
• If
{ ( )} ( ),
m
S ft Kv
= then { ( )} ( ) (0) .
mn
v
v
S f t vK v f
éù
¢=-
êú
ëû
Proof : From KAJ transform definition:
{ ( )}
m
S ft
=
()Kv =
0
1dt,
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø
ò
then
{ ( )}
m
S ft
¢
=0
1dt.
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø
¢
ò
Using integration by parts method for:
{ ( )}
m
S ft
¢=0
1dt,
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø
¢
òLet
,,
tt
u e du e
--
= =-
,,
tt
vv
dv f v vf
æö æö
ç÷ ç÷
èø èø
=
¢
= Then,
{ ( )}
m
S ft
¢
=0
1
(|
t
n
t
v
vve f
-¥
æö
ç÷
èø
é
ê
ë
0dt ,
t
t
t
v
vf e
¥-
=
æö
ç÷
èø ù
+ú
û
ò
{ ( )}
m
S ft
¢
=
1
(0 (0))
n
vvf
-+
()vK v =
( ) (0).
n
v
v
vK v f-
• If
{ ( )} ( ),
m
S ft Kv
= then
{ ( )}
m
S ft
¢¢
=2
1
[ (0)
n
vvf
--
2
(0)] ( ).vf v K v+
¢
Proof: From KAJ transform definition:
{ ( )}
m
S ft =
()Kv =
0
1dt,
t
nt
t
v
vfe
¥-
=
æö
ç÷
èø
ò
then
{ ( )}
m
S ft
¢¢ =0
1dt.
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø
¢¢
ò
Using integration by parts method for:
{ ( )}
m
S ft
¢¢
=0
1dt,
t
nt
t
v
v
fe
¥-
=
æö
ç÷
èø
¢¢
ò
Let
,,
tt
u e du e
--
= =- ,
t
v
dv f
æö
ç÷
èø
¢
=,
t
v
v vf
æö
ç÷
èø
= Then,
{ ( )}
m
S ft
¢¢
=
2
1
[ (0)
n
vvf
--
2
(0)] ( ).vf v K v+
¢

NEW INTEGRAL KAJ TRANSFORM 5
4. KAJ Integral Transform in Solving Some Differential Problems
KAJ integral transform is going to be used to solve some ordinary
differential equations, to demonstrate the transform efficiency in solving
such problems, that have important applications in scientific fields.
Problem (A) : KAJ integral transform is going to be used to solve the
following second order differential equation, given by:
1,yy
¢¢ -=
(0)y=
(0) 0y¢=
. (1)¼
Using KAJ transform on both sides of equation (1) :
][
m
Sy
¢¢
-
[ ] [1],
mm
Sy S
=2
1
[ (0)
n
vvfÞ- 2
(0)] ( )vf v K v-+
¢1
1
() ,
n
v
Kv +
-= 21( )vÞ -
1
1
() ,
n
v
Kv +
= then
[ ( )]
m
S yt
=
()Kv =
12
1
( 1)
n
vv
+-=
1
( 1)( 1) .
n
vvv v
-+
Using
partial fraction and inverse KAJ transform : 11
( 1)( 1)
() ,
mn
vvv v
yt S
-
éù
êú
êú
-+
ëû
=
1
2
() [ 2] cosh() 1.
tt
yt e e t
-
= + -= -
The concluded result is the exact
required solution for equation (1).
Problem (B) : It is possible to solve the differential equation of the nuclear
physics decay problem [10], using KAJ integral transform, as follows:
Considering the first order radioactive decay equation:
dN
dt =
( ),Nt
a
-
(2).¼
Where N(t) is the number of undecayed atoms remaining in a
sample of radioactive isotope at the time t, and
a
is the decay constant.
Applying KAJ transform to equation (2) :
( )} { ( )} 0{ ,
mm
SNt SNt
a
+=
¢
then :
()vK v -
(0)
n
v
vN+
( ) 0,Kv
a
=
()vK vÞ-
0
n
v
vN+
( ) 0,Kv
a
=
()v
a
Þ-
0
() .
n
v
v
Kv N=
Here,
{ ( )}
m
S Nt
()Kv N==
and N(0) = N0. Then,
()Kv N==
0
()
n
vN
vv
a
+=
0.
n
Nv
v
v
a
éù
êú
+
ëû
Now, taking inverse KAJ integral transform to both sides of
previous equation : 0
() .
t
Nt Ne
a
-
= The concluded result is the correct form
for radioactive decay.
5. Results and Conclusions
“Kuffi-Abbas-Jawad”KAJ integral transform and its properties
are defined and proven, the transform effectiveness in solving ordinary
differential equations have been demonstrated through the solving of

6 E. S. ABBAS, E. A. KUFFI AND A. A. JAWAD
second-order example of ordinary differential equations, and a real-life
engineering ordinary differential equation problem.
KAJ integral transform is a modification on the SEE integral
transform, the function f in KAJ integral transform depends on two
variables (t and v), while in SEE transform it depends only on the variable
t; the dependency of the function f in the proposed transform on two
variables gave it an excellent efficiency in handling special cases in signal
processing, where the signal power has a decreasing tail. KAJ integral
transform is capable to solve linear and nonlinear ordinary differential
equations, the discussed problems have proven the transform efficiency
in providing a simple solution method to some ordinary differential
equations, bytransferringthem into fundamental algebraic equations that
can solvedbyexploitingthepropertiesof thetransform.
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Received October, 2021
Revised December, 2021