# On aq-Analogue of the Elzaki Transform Called Mangontarum q-Transform

Authors: Mahid M. Mangontarum
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

Twoq-analogues of the Elzaki transform, called Mangontarum q-transforms, are introduced in this paper. Properties such as the transforms of q-trigonometric functions, transform of q-derivatives, duality relation, convolution identity, q-derivative of transforms, and transform of the Heaviside function are derived and presented. Moreover, we will consider applications of the Mangontarum q-transform of the first kind to some ordinary q-differential equations with initial values.


---

## Body

## 1. Introduction

Given the set(1)
S
=
{
f
(
t
)
:
∃
M
,
k
1
,
k
2
>
0
,
|
f
(
t
)
|
<
M
e
|
t
|
/
k
j
,
l
l
l
l
l
t
∈
(
-
1
)
j
×
[
0
,
∞
)
}
.
Elzaki [1] introduced a new integral transform calledElzaki transform defined by
(2)
E
{
f
(
t
)
}
=
T
(
v
)
=
v
∫
0
∞
‍
e
-
t
/
v
f
(
t
)
d
t
,
for f
(
t
)
∈
S, k
1
≤
v
≤
k
2, and t
≥
0. This new transform appears to be a convenient tool in solving various problems related to differential equations as seen in [1–3] and the references cited therein. Fundamental properties of this transform were already established by Elzaki et al. in [1–3] and the references therein. Among these properties are the following: (i)
the Elzaki transform of special functions such asf
(
t
)
=
t
n given by
(3)
E
{
t
n
}
=
n
!
v
n
+
2
;
(ii)
the Elzaki transform off
(
t
)
=
∑
n
≥
0
‍
a
n
t
n, given by
(4)
E
{
f
(
t
)
}
=
∑
n
≥
0
‍
n
!
a
n
v
n
+
2
;
(iii)
the Elzaki transform of derivatives given by(5)
E
{
f
(
n
)
}
=
T
(
v
)
v
n
-
∑
j
=
0
n
-
1
‍
v
2
-
n
+
j
f
(
n
)
(
0
)
;
(iv)
the duality relation given by(6)
T
(
v
)
=
v
F
(
1
v
)
,
where F
(
v
) is theLaplace transform of f
(
t
); and
(v)
the convolution identity given by(7)
E
{
(
f
*
g
)
(
t
)
}
=
1
v
M
(
v
)
N
(
v
)
,
where M
(
v
) and N
(
v
) are the Elzaki transforms of f
(
t
) and g
(
t
), respectively, and
(8)
(
f
*
g
)
(
t
)
=
∫
0
∞
‍
f
(
t
)
g
(
t
-
τ
)
d
τ
is the convolution of f
(
t
) and g
(
t
).The study of theq-analogues of classical identities is a popular topic among mathematicians and physicists. One may see [4–8] and the references therein for a brief overview and examples of q-analogues and their applications. The parameter q is known to stand for “quantum” which is widely seen inquantum calculus (or
q
-calculus). The book of Kac and Cheung [9] is a good source for further details of quantum calculus. Basically, the term “q-analogue” refers to a mathematical expression involving a parameter q which generalizes a known identity and reduces back to the known identity when q
→
1. That is, a polynomial a
k
(
q
) is said to be a q-analogue of an integer a
k if by taking its limit as q tends to 1, we recover a
k. For example, for any integers n and k with 0
<
q
<
1 and q
≠
1, we have the following q-analogues of the integer n, falling factorial (
n
)
k, factorial n
!, and binomial coefficient (
n
k
), respectively:
(9)
[
n
]
q
=
q
n
-
1
q
-
1
,
[
n
]
k
,
q
=
∏
j
=
0
k
-
1
‍
q
n
-
j
-
1
q
-
1
,
[
n
]
q
!
=
∏
j
=
1
n
‍
[
j
]
q
,
[
n
k
]
q
=
∏
j
=
1
k
‍
q
n
-
j
+
1
-
1
q
j
-
1
.
Clearly,
(10)
lim
⁡
q
→
1
[
n
]
q
=
n
,
lim
⁡
q
→
1
[
n
]
k
,
q
=
(
n
)
k
,
lim
⁡
q
→
1
[
n
]
q
!
=
n
!
,
lim
⁡
q
→
1
[
n
k
]
q
=
(
n
k
)
.
The expressions in (9) are called q
-integer, q
-falling factorial of
n
of order
k, q
-factorial of
n, and q
-binomial coefficient (orGaussian polynomial), respectively. Furthermore, the q-binomial coefficient can be expressed as
(11)
[
n
k
]
q
=
[
n
]
k
,
q
[
k
]
q
!
=
[
n
]
q
!
[
k
]
q
!
[
n
-
k
]
q
!
.
Note that the transition of any classical expression to its q-analogue is not unique. For instance, the two q-analogues for the exponential function e
t denoted by e
^
q
(
t
) and e
q
(
t
) are given, respectively, by
(12)
e
^
q
(
t
)
=
∏
i
=
1
∞
‍
(
1
+
(
1
-
q
)
q
i
-
1
t
)
=
∑
k
=
0
∞
‍
q
(
k
2
)
t
k
[
k
]
q
!
,
(13)
e
q
(
t
)
=
∏
i
=
1
∞
‍
(
1
-
(
1
-
q
)
q
i
-
1
t
)
-
1
=
∑
k
=
0
∞
‍
t
k
[
k
]
q
!
,
with e
q
(
t
)
e
^
q
(
-
t
)
=
1. Other important tools in this sequel are theJackson
q
-derivative
(14)
D
q
(
f
(
x
)
)
=
∂
q
∂
q
x
f
(
x
)
=
f
(
x
)
-
f
(
q
x
)
(
1
-
q
)
x
and thedefinite Jackson
q
-integral
(15)
∫
0
x
‍
f
(
t
)
d
q
t
=
(
1
-
q
)
∑
i
≥
0
‍
f
(
q
i
x
)
x
q
i
.
Note that from (14) and (15), we have
(16)
∫
0
x
‍
∂
q
∂
q
t
f
(
t
)
d
q
t
=
f
(
x
)
-
f
(
0
)
.
Furthermore, given the improper q-integral of f
(17)
∫
0
∞
/
a
‍
f
(
x
)
d
q
x
=
(
1
-
q
)
∑
n
∈
Z
‍
q
n
a
f
(
q
n
a
)
,
we get (see [10])
(18)
∫
x
∞
‍
f
(
x
)
d
q
x
=
∫
0
x
·
∞
‍
f
(
x
)
d
q
x
-
∫
0
∞
‍
f
(
x
)
d
q
x
.Forq-analogues of integral transforms, Chung and Kim [10] defined the q-Laplace transform of the first and second kind given, respectively, by
(19)
F
q
(
s
)
=
L
q
{
f
(
t
)
}
=
∫
0
∞
‍
e
^
q
(
-
q
s
t
)
f
(
t
)
d
q
t
,
(20)
F
~
q
(
s
)
=
L
~
q
{
f
(
t
)
}
=
∫
0
∞
‍
e
q
(
-
s
t
)
f
(
t
)
d
q
t
,
for s
>
0. Other q-analogues of the Laplace transform were earlier considered by Hahn [11] and Abdi [12]. This motivates us to define a q-analogue for the Elzaki transform in [1].In this paper, we will define two kinds ofq-analogues of the Elzaki transform, and to differentiate them from other possible q-analogues, we will refer to these transforms asMangontarum
q
-transforms. Some properties and interesting formulae are derived and presented.


## 2. Mangontarumq-Transform of the First Kind

Let(21)
A
=
{
f
(
t
)
:
∃
M
,
k
1
,
k
2
>
0
,
|
f
(
t
)
|
<
M
e
^
q
(
|
t
|
k
j
)
,
l
l
l
l
l
l
t
∈
(
-
1
)
j
×
[
0
,
∞
)
(
|
t
|
k
j
)
}
.
Hence, we have the following definition.Definition 1.
The Mangontarumq-transform of the first kind, denoted by E
q, is defined by
(22)
E
q
{
f
(
t
)
}
=
T
q
(
v
)
=
v
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
f
(
t
)
d
q
t
,
over the set in (21), where k
1
<
v
<
k
2 and t
≥
0.Clearly,(23)
lim
⁡
q
→
1
E
q
{
f
(
t
)
}
=
E
{
f
(
t
)
}
.
This makes (22) a q-analogue of the Elzaki transform in (2). Also, from Definition 1, we can easily observe thelinearity relation
(24)
E
q
{
α
f
(
t
)
+
β
g
(
t
)
}
=
α
E
q
{
f
(
t
)
}
+
β
E
q
{
g
(
t
)
}
,
where f
(
t
), g
(
t
)
∈
A, and α, β are real numbers.Aq-analogue of thegamma function
(25)
Γ
(
t
)
=
∫
0
∞
‍
e
-
x
x
t
-
1
d
x
,
for t a complex number, is given by
(26)
Γ
q
(
t
)
=
∫
0
∞
‍
x
t
-
1
e
^
q
(
-
q
x
)
d
q
x
,
for t
>
0 and
(27)
Γ
q
(
t
+
1
)
=
[
t
]
q
Γ
(
t
)
,
Γ
q
(
n
+
1
)
=
[
n
]
q
!
,
where n is a positive integer (see [10, 13, 14]).Since (22) can be expressed as
(28)
T
q
(
v
)
=
v
2
∫
0
∞
‍
e
^
q
(
-
q
t
)
f
(
v
t
)
d
q
t
,
then by (28) and (27), we have the following:
(29)
E
q
{
1
}
=
v
2
,
E
q
{
t
}
=
v
3
,
E
q
{
t
n
}
=
v
n
+
2
[
n
]
q
!
.
Moreover, by (24) and (29), we have
(30)
E
q
{
g
(
t
)
}
=
∑
n
≥
0
‍
a
n
v
n
+
2
[
n
]
q
!
,
E
q
{
e
q
(
a
t
)
}
=
v
2
1
-
a
v
,
E
q
{
e
^
q
(
a
t
)
}
=
v
2
∑
n
≥
0
‍
q
(
n
2
)
(
a
v
)
n
,
where g
(
t
)
=
∑
n
≥
0
‍
a
n
t
n.Given theq-exponential function
(31)
e
q
(
i
t
)
=
∑
n
≥
0
‍
i
n
t
n
[
n
]
q
!
=
∑
n
≥
0
‍
(
-
1
)
n
[
2
n
]
q
!
t
2
n
+
i
∑
n
≥
0
‍
(
-
1
)
n
[
2
n
+
1
]
q
!
t
2
n
+
1
,
the q-sine and q-cosine functions can be defined as
(32)
cos
⁡
q
(
z
)
=
∑
n
≥
0
‍
(
-
1
)
n
[
2
n
]
q
!
z
2
n
=
1
2
(
e
q
(
i
z
)
+
e
q
(
-
i
z
)
)
,
sin
q
(
z
)
=
∑
n
≥
0
‍
(
-
1
)
n
[
2
n
+
1
]
q
!
z
2
n
+
1
=
1
2
i
(
e
q
(
i
z
)
-
e
q
(
-
i
z
)
)
,
where i
=
-
1. Now, by (28) and (27),
(33)
E
q
{
cos
⁡
q
(
t
)
}
=
v
2
∑
n
≥
0
‍
(
-
1
)
n
v
2
n
[
2
n
]
q
!
∫
0
∞
‍
t
2
n
e
^
q
(
-
q
t
)
d
q
t
=
v
2
∑
n
≥
0
‍
(
-
1
)
n
v
2
n
[
2
n
]
q
!
Γ
q
(
2
n
+
1
)
=
v
2
∑
n
≥
0
‍
(
-
v
2
)
n
=
v
2
1
+
v
2
.
Similarly,
(34)
E
q
{
sin
q
(
t
)
}
v
3
1
+
v
2
.
Now, if we define thehyperbolic
q-sineand
q-cosine functions as
(35)
cosh
⁡
q
(
t
)
=
e
q
(
t
)
+
e
q
(
-
t
)
2
,
sinh
⁡
q
(
t
)
=
e
q
(
t
)
-
e
q
(
-
t
)
2
,
we have the next theorem which presents the transforms of the q-trigonometric functions.Theorem 2 (transform ofq-trigonometric functions).
The following identities hold:(36)
E
q
{
cos
⁡
q
(
a
t
)
}
=
v
2
1
+
a
2
v
2
;
E
q
{
sin
q
(
a
t
)
}
=
a
v
3
1
+
a
2
v
2
;
E
q
{
cosh
⁡
q
(
a
t
)
}
=
v
2
1
-
a
2
v
2
;
E
q
{
sinh
⁡
q
(
a
t
)
}
=
a
v
3
1
-
a
2
v
2
.Proof.
E
q
{
cos
⁡
q
(
a
t
)
} and E
q
{
sin
q
(
a
t
)
} can be proved parallel to (33). For E
q
{
cosh
⁡
q
(
a
t
)
}, we have
(37)
E
q
{
cosh
⁡
q
(
a
t
)
}
=
1
2
[
E
q
{
e
q
(
a
t
)
}
+
E
q
{
e
q
(
-
a
t
)
}
]
=
1
2
[
v
2
1
-
a
v
+
v
2
1
+
a
v
]
=
v
2
1
-
a
2
v
2
.
E
q
{
sinh
⁡
q
(
a
t
)
} can be proven similarly.The knownq-product rule of differentiation is given by
(38)
D
q
(
f
(
t
)
g
(
t
)
)
=
D
q
(
f
(
t
)
)
g
(
t
)
+
f
(
q
t
)
D
q
(
g
(
t
)
)
.
Thus, we have
(39)
∫
0
∞
‍
D
q
(
f
(
t
)
g
(
t
)
)
d
q
=
∫
0
∞
‍
D
q
(
f
(
t
)
)
g
(
t
)
d
q
+
∫
0
∞
‍
f
(
q
t
)
D
q
(
g
(
t
)
)
d
q
.
Let f
(
t
)
=
e
^
q
(
-
t
/
v
) so that f
(
q
t
)
=
e
^
q
(
-
q
(
t
/
v
)
). Since
(40)
D
q
(
e
^
q
(
-
q
t
v
)
)
=
-
1
v
e
^
q
(
-
q
t
v
)
,
then from (39), we have
(41)
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
D
q
(
g
(
t
)
)
d
q
=
-
g
(
0
)
+
1
v
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
g
(
t
)
d
q
.
Thus, from Definition 1,
(42)
E
q
{
D
q
(
g
(
t
)
)
}
=
-
v
g
(
0
)
+
E
q
{
g
(
t
)
}
v
.LetD
q
n
(
f
(
t
)
) denote the nth q-derivative of the function f
(
t
)
∈
A. Then, we have the following theorem.Theorem 3 (transform ofq-derivatives).
Iff
(
t
), D
q
(
f
(
t
)
)
,
…
,
D
q
n
(
f
(
t
)
)
∈
A, then
(43)
E
q
{
D
q
n
(
f
(
t
)
)
}
=
E
q
{
f
(
t
)
}
v
n
-
∑
j
=
0
n
-
1
‍
v
j
-
n
+
2
D
q
j
(
f
(
0
)
)
.Proof (by induction).
The case whenn
=
1 is justified by (42). Now, suppose (43) holds for n
=
k
>
0. That is
(44)
E
q
{
D
q
k
(
f
(
t
)
)
}
=
E
q
{
f
(
t
)
}
v
k
-
∑
j
=
0
k
-
1
‍
v
j
-
k
+
2
D
q
j
(
f
(
0
)
)
.
Let g
(
t
)
=
D
q
k
(
f
(
t
)
) so that D
q
(
g
(
t
)
)
=
D
q
k
+
1
(
f
(
t
)
). Thus, by (42) and (43) we have
(45)
E
q
{
D
q
k
+
1
(
f
(
t
)
)
}
=
E
q
{
D
q
(
g
(
t
)
)
}
=
E
q
{
g
(
t
)
}
v
-
v
g
(
0
)
=
E
q
{
f
(
t
)
}
v
k
+
1
-
∑
j
=
0
k
-
1
‍
v
j
-
k
+
1
D
q
j
(
f
(
0
)
)
-
v
D
q
k
(
f
(
0
)
)
=
E
q
{
f
(
t
)
}
v
k
+
1
-
∑
j
=
0
k
‍
v
j
-
k
+
1
D
q
j
(
f
(
0
)
)
.
Hence, the proof is done.LetF
q
(
s
) be the q-Laplace transform of the first kind. Replacing s with 1
/
v in (19) yields
(46)
F
q
(
1
v
)
=
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
f
(
t
)
d
q
t
.
The next theorem is obtained by multiplying both sides of this equation by v.Theorem 4 (duality relation).
LetF
q
(
s
) be the q-Laplace transform of the first kind of f
(
t
)
∈
A. Then the Mangontarum q-transform of the first kind satisfies the relation
(47)
T
q
(
v
)
=
v
F
q
(
1
v
)
.Remark 5.
Note that whenv
=
1 and s
=
1, T
q
(
1
)
=
F
q
(
1
).Forf
(
t
)
=
t
α and g
(
t
)
=
t
β
-
1 (α, β
>
0), the q
-convolution of f and g is defined in [10] as
(48)
(
f
*
g
)
(
t
)
=
∫
0
t
‍
f
(
τ
)
g
(
t
-
q
τ
)
d
q
τ
,
where
(49)
g
(
t
-
q
τ
)
=
∏
i
=
0
β
-
2
‍
(
t
-
q
i
τ
)
=
∑
k
=
0
β
-
1
‍
[
β
-
1
k
]
q
(
-
1
)
k
q
(
k
2
)
τ
k
is the q
-binomial theorem. If f
(
t
)
=
∑
i
≥
0
‍
a
i
t
i
α, then theconvolution identity for the q-Laplace transform of the first kind in [10] is
(50)
L
q
{
(
f
*
g
)
(
t
)
}
=
L
q
{
f
(
t
)
}
·
L
q
{
g
(
t
)
}
.
Now, from (47) and (50),
(51)
E
q
{
(
f
*
g
)
(
t
)
}
=
v
L
q
{
(
f
*
g
)
(
t
)
}
=
v
[
F
q
(
1
v
)
·
G
q
(
1
v
)
]
=
1
v
M
q
(
v
)
·
N
q
(
v
)
.
Thus, we have the following convolution identity for the Mangontarum q-transform of the first kind.Theorem 6 (convolution identity).
Iff
(
t
), g
(
t
)
∈
A with q-Laplace transforms of the first kind F
q
(
s
) and G
q
(
s
), respectively, and Mangontarum q-transforms of the first kind M
q
(
v
) and N
q
(
v
), respectively, then
(52)
E
q
{
(
f
*
g
)
(
t
)
}
=
1
v
M
q
(
v
)
·
N
q
(
v
)
.From [10], the q-derivative of the q-Laplace transform of the first kind is
(53)
L
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
s
)
n
F
q
(
q
-
n
s
)
.
Replacing s with 1
/
v and applying Theorem 6 yield
(54)
E
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
v
∫
0
∞
‍
e
^
q
(
-
q
t
v
q
n
)
f
(
t
)
d
q
t
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
q
(
v
q
n
)
.
Hence, the following theorem is easily observed.Theorem 7 (q-derivative of transforms).
Forn a positive integer and f
(
t
)
∈
A, one has
(55)
E
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
q
(
v
q
n
)
.The next corollary follows directly from (55).Corollary 8.
Forn a positive integer and f
(
t
)
∈
A,
(56)
E
q
{
e
q
(
a
t
)
f
(
t
)
}
=
∑
n
≥
0
‍
(
-
a
)
n
[
n
]
q
!
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
q
(
v
q
n
)
,
E
q
{
e
q
(
a
t
)
t
n
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
v
2
q
n
1
-
a
v
q
n
.TheHeaviside function
H
(
t
-
a
) is defined by
(57)
H
(
t
-
a
)
=
{
1
,
if
t
≥
a
,
0
,
if
0
≤
t
≤
a
.
Now, we have the following theorem.Theorem 9.
LetH
(
t
-
a
) be the Heaviside function. Then, one has
(58)
E
q
{
H
(
t
-
a
)
}
=
v
e
^
q
(
-
a
v
)
.Proof.
From Definition1 and (18),
(59)
E
q
{
H
(
t
-
a
)
}
=
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
H
(
t
-
a
)
d
q
t
=
∫
a
∞
‍
e
^
q
(
-
q
t
v
)
d
q
t
=
∫
0
∞
‍
e
^
q
(
-
q
t
v
)
d
q
t
-
∫
0
a
‍
e
^
q
(
-
q
t
v
)
d
q
t
.
Since E
q
{
1
}
=
v
2 and by (12),
(60)
E
q
{
H
(
t
-
a
)
}
=
v
-
∑
n
≥
0
‍
(
-
1
)
n
q
(
n
2
)
(
q
/
v
)
n
[
n
]
q
!
∫
0
a
‍
t
n
d
q
t
=
v
-
∑
n
≥
0
‍
(
-
1
)
n
q
(
n
+
1
2
)
[
n
+
1
]
q
!
v
n
a
n
+
1
.
Further simplifications lead to (58).


## 3. Application ofE
q

In this section, we will consider applications of the Mangontarumq-transform of the first kind to some q-differential equations. To obtain this, the following definition is essential.Definition 10.
Letf
(
t
)
∈
A. If E
q
{
f
(
t
)
}
=
f
(
v
), we say that f
(
t
) is an inverse Mangontarum q-transform (of the first kind), or an inverse q-transform of the function f
(
v
), and we write
(61)
f
(
t
)
=
E
q
-
1
{
f
(
s
)
}
.Observe that linearity also holds for the inverseq-transform of the function f
(
v
). That is,
(62)
E
q
-
1
{
α
f
(
s
)
+
β
g
(
s
)
}
=
α
E
q
-
1
{
f
(
s
)
}
+
β
E
q
-
1
{
g
(
s
)
}
,
where α
,
β
∈
A.Theq-difference calculus (or quantum calculus) was first studied by Jackson [15], Carmichael [16], Mason [17], Adams [18], and Trjitzinsky [19] in the early 20th century. Some recent studies involving q-differential equations are those by Ahmad et al. [20], Yu and Wang [21], and Sitthiwirattham et al. [22]. In the following examples, the effectiveness of the Mangontarum q-transform of the first kind in solving certain initial value problem involving ordinary q-differential equations is illustrated.Example 11.
Consider the first degreeq-differential equation:
(63)
D
q
(
f
(
x
)
)
+
f
(
x
)
=
0
,
where f
(
x
)
∈
A with initial condition f
(
0
)
=
1. Taking the Mangontarum q-transform (of the first kind) of both sides of this equation yields
(64)
E
q
{
D
q
(
f
(
x
)
)
}
+
E
q
{
f
(
x
)
}
=
0
.
Let T
q
(
v
)
=
E
q
{
f
(
x
)
}. Applying Theorem 3, we have
(65)
E
q
{
D
q
(
f
(
x
)
)
}
=
T
q
(
v
)
v
-
v
.
Hence,
(66)
T
q
(
v
)
v
-
v
+
3
T
q
(
v
)
=
0
,
T
q
(
v
)
(
1
+
3
v
v
)
=
v
,
E
q
{
f
(
x
)
}
=
v
2
1
+
3
v
.
Using the inverse q-transform in (61) yields the solution
(67)
f
(
x
)
=
E
q
-
1
{
v
2
1
+
3
v
}
=
e
q
(
-
3
x
)
.Example 12.
Find the solution of the second orderq-differential equation:
(68)
∂
q
y
∂
q
x
-
y
=
0
,
with initial condition f
(
0
)
=
0 and f
1
(
0
)
=
5.
Solution. If T
q
(
v
)
=
E
q
{
y
}, then by Theorem 3
(69)
E
q
{
∂
q
y
∂
q
x
}
=
T
q
(
v
)
v
2
-
5
v
.
Thus, taking the Mangontarum q-transform (of the first kind) of both sides of (68) gives us
(70)
T
q
(
v
)
v
2
-
5
v
-
T
q
(
v
)
=
0
,
T
q
(
v
)
(
1
-
v
2
v
2
)
=
5
v
,
E
q
{
y
}
=
5
v
3
1
-
v
2
.
From the inverse q-transform in (61), we have the solution
(71)
y
=
E
q
-
1
{
5
v
3
1
-
v
2
}
=
5
sinh
⁡
q
(
x
)
.Example 13.
Find the solution of the equation(72)
y
′′
-
3
y
′
+
2
y
=
1
,
where y
′′
=
(
∂
q
/
∂
q
x
)
2
y and y
′
=
∂
q
y
/
∂
q
x with y
(
0
)
=
0 and y
′
(
0
)
=
1.
Solution. Let T
q
(
v
)
=
E
q
{
y
}. Since
(73)
E
q
{
(
∂
q
∂
q
x
)
2
y
}
=
T
q
(
v
)
v
2
,
E
q
{
∂
q
y
∂
q
x
}
=
T
q
(
v
)
v
,
then the Mangontarum q-transform of (72) yields
(74)
T
q
(
v
)
v
2
-
3
T
q
(
v
)
v
2
+
2
T
q
(
v
)
=
v
2
,
T
q
(
v
)
(
1
-
3
v
+
2
v
2
v
2
)
=
v
2
,
T
q
(
v
)
=
v
3
2
v
2
-
3
v
+
1
,
E
q
{
y
}
=
-
v
2
1
-
v
+
2
v
2
1
-
2
v
.
Applying (61) yields the solution
(75)
y
=
-
e
q
(
x
)
+
2
e
q
(
2
x
)
.Example 14.
Find the solution of(76)
y
′′
+
y
=
6
sin
q
(
2
x
)
when y
(
0
)
=
3 and y
′′
(
0
)
=
1.
Solution. Let T
q
(
v
)
=
E
q
{
y
}. To solve (76), we make use of the following results:
(77)
E
q
{
(
∂
q
∂
q
x
)
2
y
}
=
T
q
(
v
)
v
2
-
3
-
v
,
E
q
{
sin
q
(
2
x
)
}
=
2
v
3
1
+
4
v
2
.
Hence, the q-transform of (76) is
(78)
E
q
{
y
′′
}
+
E
q
{
y
}
-
E
q
{
6
sin
q
(
2
x
)
}
=
T
q
(
v
)
v
2
-
3
-
v
+
T
q
(
v
)
-
12
v
3
1
+
4
v
2
.
Further simplifications yield
(79)
T
q
(
v
)
=
3
v
2
1
+
v
2
+
5
v
3
1
+
v
2
-
4
v
3
1
+
4
v
2
.
The inverse q-transform in (61) gives the solution
(80)
y
=
3
cos
⁡
q
(
x
)
+
5
sin
q
(
x
)
-
2
sin
q
(
2
x
)
.


## 4. Mangontarumq-Transform of the Second Kind

Since there can be more than oneq-analogue of any classical expression, we can define another q-analogue for the Elzaki transform. Let
(81)
A
-
=
{
f
(
t
)
:
∃
M
,
k
1
,
k
2
>
0
,
|
f
(
t
)
|
<
M
e
q
(
|
t
|
k
j
)
,
l
l
l
l
l
l
t
∈
(
-
1
)
j
×
[
0
,
∞
)
(
|
t
|
k
j
)
}
.
Then, we have the following definition.Definition 15.
The Mangontarumq-transform of the second kind, denoted by E
~
q, is defined by
(82)
E
~
q
{
f
(
t
)
}
=
T
~
q
(
v
)
=
v
∫
0
∞
‍
e
q
(
-
t
v
)
f
(
t
)
d
q
t
,
over the set A
-, where k
1
<
v
<
k
2 and t
≥
0.Clearly,lim
⁡
q
→
1
E
~
q
{
f
(
t
)
}
=
E
{
f
(
t
)
}. Moreover, it is easily observed that
(83)
E
~
q
{
α
f
(
t
)
+
β
g
(
t
)
}
=
α
E
~
q
{
f
(
t
)
}
+
β
E
~
q
{
g
(
t
)
}
,
where f
(
t
), g
(
t
)
∈
A
-, and α, β are real numbers.Given theq-gamma function of the second kind
(84)
γ
q
(
t
)
=
∫
0
∞
‍
x
t
-
1
e
q
(
-
x
)
d
q
x
,
for t
>
0, we get
(85)
γ
q
(
1
)
=
1
,
γ
q
(
t
+
1
)
=
q
-
t
[
t
]
q
γ
q
(
t
)
,
γ
q
(
n
)
=
q
-
(
n
2
)
Γ
q
(
n
)
,
for n a positive integer. Note that (82) may be expressed as
(86)
T
~
q
(
v
)
=
v
2
∫
0
∞
‍
e
q
(
-
t
)
f
(
v
t
)
d
q
t
.
Hence, from (86), we have the following results:
(87)
E
~
q
{
1
}
=
v
2
,
E
~
q
{
t
}
=
v
3
q
,
E
~
q
{
t
n
}
=
v
n
+
2
q
(
n
+
1
2
)
[
n
]
q
!
,
E
~
q
{
∑
n
≥
0
‍
a
n
t
n
}
=
∑
n
≥
0
‍
a
n
v
n
+
2
q
(
n
+
1
2
)
[
n
]
q
!
,
E
~
q
{
e
q
(
a
t
)
}
=
∑
n
≥
0
‍
a
n
v
n
+
2
q
(
n
+
1
2
)
,
E
~
q
{
e
^
q
(
a
t
)
}
=
v
2
q
q
-
a
v
.From (12),
(88)
e
^
q
(
i
t
)
=
∑
n
≥
0
‍
(
-
1
)
n
q
(
2
n
2
)
[
2
n
]
q
!
t
2
n
+
i
∑
n
≥
0
‍
(
-
1
)
n
q
(
2
n
+
1
2
)
[
2
n
+
1
]
q
!
t
2
n
+
1
.
Hence, new q-analogues of the sine and cosine functions can be defined, respectively, as (see [10])
(89)
cos
⁡
q
′
(
a
t
)
=
∑
n
≥
0
‍
(
-
1
)
n
q
(
2
n
2
)
[
2
n
]
q
!
(
a
t
)
2
n
=
1
2
(
e
^
q
(
i
a
t
)
+
e
^
q
(
-
i
a
t
)
)
;
sin
q
′
(
a
t
)
=
∑
n
≥
0
‍
(
-
1
)
n
q
(
2
n
+
1
2
)
[
2
n
+
1
]
q
!
(
a
t
)
2
n
+
1
=
1
2
i
(
e
^
q
(
i
a
t
)
-
e
^
q
(
-
i
a
t
)
)
,
and the new q-hyperbolic sine and cosine functions as
(90)
cosh
⁡
q
′
(
a
t
)
=
1
2
(
e
^
q
(
a
t
)
+
e
^
q
(
-
a
t
)
)
;
sinh
⁡
q
′
(
a
t
)
=
1
2
(
e
^
q
(
a
t
)
-
e
^
q
(
-
a
t
)
)
.
By application of E
~
q
{
t
n
}
=
(
v
n
+
2
/
q
(
n
+
1
2
)
)
[
n
]
q
!, it is easy to obtain the next theorem.Theorem 16 (transform ofq-trigonometric functions).
The following statements are true:(91)
E
~
q
{
cos
⁡
q
′
(
a
t
)
}
=
v
2
q
2
q
2
+
a
2
v
2
;
E
~
q
{
sin
q
′
(
a
t
)
}
=
a
q
v
3
q
2
+
a
2
v
2
;
E
~
q
{
cosh
⁡
q
′
(
a
t
)
}
=
v
2
q
2
q
2
-
a
2
v
2
;
E
~
q
{
sinh
⁡
q
′
(
a
t
)
}
=
a
q
v
3
q
2
-
a
2
v
2
.From (20),
(92)
F
~
q
(
s
)
=
∫
0
∞
‍
e
q
(
-
s
t
)
f
(
t
)
d
q
t
.
Replacing s with 1
/
v and multiplying both sides with v yield
(93)
v
F
~
q
(
1
v
)
=
T
~
q
(
v
)
.
Thus, we have the duality relation between the q-Laplace transform of the second kind and the Mangontarum q-transform of the second kind as stated in the next theorem.Theorem 17 (duality relation).
LetF
~
q
(
s
) be the q-Laplace transform of the second kind of f
(
t
)
∈
A. Then the Mangontarum q-transform of the second kind satisfies the relation
(94)
T
~
q
(
v
)
=
v
F
~
q
(
1
v
)
.From [10], for any positive integer n,
(95)
L
~
q
{
f
(
n
)
(
t
)
}
=
s
n
q
-
(
n
+
1
2
)
F
~
q
(
q
-
n
s
)
-
∑
i
=
0
n
-
1
‍
s
n
-
1
-
i
q
-
(
n
-
i
2
)
f
(
i
)
(
0
)
,
where F
~
q
(
q
-
n
s
)
=
∫
0
∞
‍
e
q
(
-
q
-
n
s
t
)
f
(
t
)
d
q, f
(
n
)
(
t
)
=
(
∂
q
/
∂
q
t
)
n
f
(
t
). For f
(
t
)
∈
A
-, let G
~
q
(
s
)
=
L
~
q
{
f
(
n
)
(
t
)
} and V
~
q
(
v
)
=
E
~
q
{
f
(
n
)
(
t
)
}. Then, by Theorem 17,
(96)
V
~
q
(
v
)
=
v
G
~
q
(
1
v
)
=
1
v
n
q
-
(
n
+
1
2
)
v
∫
0
∞
‍
e
q
(
-
t
q
n
v
)
-
∑
i
=
0
n
-
1
‍
v
i
-
n
+
2
q
-
(
n
-
i
2
)
f
(
i
)
(
0
)
=
q
-
(
n
+
1
2
)
T
~
q
(
q
n
v
)
v
n
-
∑
i
=
0
n
-
1
‍
v
i
-
n
+
2
q
-
(
n
-
i
2
)
D
q
i
(
f
(
0
)
)
.
Hence, the following theorem is obtained.Theorem 18 (transform ofq-derivatives).
Iff
(
t
), D
q
(
f
(
t
)
)
,
…
,
D
q
n
(
f
(
t
)
)
∈
A
-, then
(97)
E
~
q
{
D
q
n
f
(
t
)
}
=
q
-
(
n
+
1
2
)
T
~
q
(
q
n
v
)
v
n
-
∑
i
=
0
n
-
1
‍
v
i
-
n
+
2
q
-
(
n
-
i
2
)
D
q
i
(
f
(
0
)
)
.Theq-derivative of the q-Laplace transform of the second kind [10] is
(98)
L
~
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
s
)
n
F
~
q
(
s
)
.
Replacing s with 1
/
v and applying Theorem 6 yield
(99)
E
~
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
v
∫
0
∞
‍
e
q
(
-
q
t
v
)
f
(
t
)
d
q
t
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
~
q
(
v
)
.
Thus, the next theorem holds.Theorem 19 (q-derivative of transforms).
Forn a positive integer and f
(
t
)
∈
A
-, one has
(100)
E
~
q
{
t
n
f
(
t
)
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
~
q
(
v
)
.Moreover, we have the following corollary.Corollary 20.
Forn a positive integer and f
(
t
)
∈
A
-,
(101)
E
~
q
{
e
^
q
(
a
t
)
f
(
t
)
}
=
∑
n
≥
0
‍
(
-
a
)
n
[
n
]
q
!
q
(
n
2
)
(
∂
q
∂
q
v
)
n
T
~
q
(
v
)
,
E
~
q
{
e
^
q
(
a
t
)
t
n
}
=
(
-
1
)
n
q
(
n
2
)
(
∂
q
∂
q
v
)
n
v
2
q
q
-
a
v
.Much is yet to be discovered regarding the Mangontarumq-transforms. The possibility of deriving more properties for these q-transforms is an interesting recommendation. One may read [1–3] and the references therein for guidance regarding this matter. Also, note that there were no applications shown for the Mangontarum q-transform of the second kind in this paper. Researchers are encouraged to further investigate other applications of these q-transforms, especially the second kind.



---

*Source: 825618-2014-10-13.xml*