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R E S E A R C H

Open Access

Certain results related to the

N

-transform

of a certain class of functions and differential

operators

Shrideh K Al-Omari

*

*Correspondence:

s.k.q.alomari@fet.edu.jo Department of Physics and Basic Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan

Abstract

In this paper, we aim to investigateq-analogues of the natural transform on various elementary functions of special type. We obtain results associated with classes of

q-convolution products, Heaviside functions,q-exponential functions,q-hyperbolic functions andq-trigonometric functions as well. Further, we give definitions and derive results involving someq-differential operators.

Keywords: q-differential operator;q-trigonometric function;q-convolution product;

q-natural transform;q-hyperbolic function; Heaviside function

1 Introduction

The subject of fractional calculus has gained noticeable importance and popularity due to its mainly demonstrated applications in diverse fields of science and engineering prob-lems. Much of developed theories of fractional calculus were based upon the familiar Riemann-Liouville activity related to theq-calculus has been noticed due to applications of this theory in mathematics, statistics and physics.

Jackson in [1] presented a precise definition and developed the q-calculus in a sys-tematic way. Thereafter, some remarkable integral transforms were associated with dif-ferent analogues in aq-calculus context. Among those examined integrals, we recall the

q-Laplace integral operator [2–5], theq-Sumudu integral operator [6–9], the Weyl frac-tionalq-integral operator [10], theq-wavelet integral operator [11], theq-Mellin type in-tegral operator [12], Mangontarum transform [13], Widder potential transform [14],E2;1

-transform [15] and a few others. In this paper, we give someq-analogues of some recently investigated integral transform, named the natural transform, and obtain some new de-siredq-properties.

In the following section, we present some notations and terminologies from the q -calculus. In Section 3, we give the definition and properties of the natural transform. In Section 4, we give the definition of the analogueNqof the natural transform and apply it

to some functions of special type. In more details, Sections 5 to 7 are associated with the application of the analogueNqto Heaviside functions, convolutions and differentiations.

The remaining two sections investigate theNqanalogue of the natural transform on some elementary functions with similar approach.

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2 Definitions and preliminaries

We present some usual notions and notations used in theq-calculus [16–18]. Throughout this paper, we assumeqto be a fixed number satisfying 0 <q< 1. Theq-calculus begins with the definition of theq-analoguedqf(x) of the differential of functions

dqf(x) =f(qx) –f(x). (1)

Having said this, we immediately get theq-analogue of the derivative off(x), called its

q-derivative,

(Dqf)(x) :=

dqf(x)

dqx

:=f(x) –f(qx)

(1 –q)x , x= 0, (2)

(Dqf)(0) =f´(0) providedf´(0) exists. Iff is differentiable, then (Dqf)(x) tends tof´(0) asq

tends to 1. It can be seen that theq-derivative satisfies the followingq-analogue of the Leibniz rule:

Dq

f(x)g(x)=g(x)Dqf(x) +f(qx)Dqg(x). (3)

Theq-Jackson integrals are defined by [1, 12]

x

0

f(x)dqx= (1 –q)x

0

fxqkqk, (4)

0

f(x)dqx= (1 –q)x

–∞

fqkqk, (5)

provided the sum converges absolutely.

Theq-Jackson integral in a generic interval [a,b] is given by [1]

b

a

f(x)dqx=

b

0

f(x)dqx

a

0

f(x)dqx. (6)

The improper integral is defined in the way that

A

0

f(x)dqx= (1 –q) ∞

–∞ f

qk

A

qk

A, (7)

and, fornZ, we have

qn

0

f(x)dqx=

0

f(x)dqx. (8)

Theq-integration by parts is defined for functionsf andgby

b

0

g(x)Dqf(x)dqx=f(b)g(b) –f(a)g(a) –

b

a

(3)

For xC, the q-shifted factorials are defined by (x;q)0= 1; (x;q)t = (xq(x;tq;q)∞); (x;q)n=

n–1

0 (1 –xqk), and hence

(x;q)∞= ∞

k=0

1 –xqk. (10)

Theq-analogues ofxand∞are respectively defined by

[x] =1 –q

x

1 –q, [∞] =

1

1 –q. (11)

Theq-analogues of the exponential function of the first and second kind are respectively given as follows:

Eq(x) = ∞

n=0

qn(n2–1) x n

([n]q)!

(xC) (12)

and

eq(x) = ∞

n=0 xn

([n]q)!

|x|< 1, (13)

where ([n]q)! = [n]q[n– 1]q· · ·[2]q[1]q, [n] = 1–q

n

1–q =q

n–1+· · ·+q+ 1.

However, in view of the product expansions, (eq(x))–1=Eq(–x), which explains the need

for both analogues of the exponential function.

The q-derivative of Eq(x) is DqEq(x) =Eq(qx), whereas the q-derivative of eq(x) is

Dqeq(x) =eq(x),eq(0) = 1.

The gamma and beta functions satisfy theq-integral representations

q(t) =

1 1–q

0 xt–1Eq(–qx)dqx

Bq(t;s) =

1

0xt–1(1 –qx)qs–1dqx (t,s> 0)

(14)

and are related to the identitiesBq(t;s) = q(s)q(t)

q(s+t) andq(t+ 1) = [t]qq(t). Due to (12)

and (13), the q-analogues of sine and cosine functions of the second and first kind are respectively given as

sinq(at) =∞0 (–1)n(at)2n+1 ([2n+1]q)!;

cosq(at) =∞0 (–1)n([2(atn)]2qn)!;

sinq(at) =∞0 (–1)n q

n(n+1) 2

([2n+1]q)!a

2n+1t2n+1;

cosq(at) =

0 (–1)n q

n(n–1) 2

([2n]q)!a

2nt2n.

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

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3 TheN-transform or the natural transform

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Maxwell’s equation in conducting media. In [21], the author applied the natural transform to some ordinary differential equations and some space of Boehmians. For further inves-tigations of the natural transform, one can refer to [22] and [23]. Over the setA, where

A={f(t)|∃τ1,τ2,M> 0,|f(t)|<Met/τj,t∈(–1)j×[0,∞)|}, theN-transform off(t) is given

as [19, (1)],

(Nf)(u;v) =

0

f(ut)exp(–vt)dt (u,v> 0). (16)

Provided the integral above exists, it is so easy to see that

(Nf)(1;v) = (Lf)(v), (Nf)(u; 1) = (Sf)(u), (17)

whereSf andLf are respectively the Sumudu and Laplace transforms off. Moreover, the natural-Laplace and natural-Sumudu dualities are given as [21, (5), (6)],

(Nf)(u;v) =

0

1

uf(t)exp

vt

u

dt (18)

and

(Nf)(u;v) =

0

1

vf

ut v

exp(–t)dt, (19)

respectively.

Further, from (18) and (19), it can be easily observed that

(Nf)(u;v) = 1

u(Lf)

u v

, (Nf)(u;v) =1

v(Sf)

u v

. (20)

The natural transform of some known functions is given as follows [21, p. 731]:

(i) N(a)(u;v) = 1v, whereais a constant; (ii) N(δ)(u;v) =1v, whereδis the delta function; (iii) N(eat)(u;v) =v1au,ais a constant.

4 TheNqanalogue of the natural transform

Hahn [24] and later Ucar and Albayrak [4] defined theq-analogues of the first and second type of Laplace transforms by means of theq-integrals

Lq(f;s) =

1 1 –q

1

s

0

Eq(qst)f(t)dqt, (21)

qL(f;s) =

1 1 –q

0

eq(–st)f(t)dqt. (22)

On the other hand, theq-analogues of the Sumudu transform of the first and second type were defined by [6, 7]

Sq(f;s) =

1 (1 –q)s

s

0 Eq

q st

(5)

qS(f;s) =

analogue of the natural transform of the first kind is

(Nqf)(u;v) :=

provided the functionf(t) is defined onAanduandvare the transform variables. The

q-series representation of (25) can be written as

(Nqf)(u;v) =

which, by (10), can be put into the form

(Nqf)(u;v) =

Let us now discuss some general applications ofNqto some elementary functions.

Theorem 1 LetαR.Then we have Proof By setting the variables, we get

The theorem hence follows. The direct consequence of (28) is that

Theorem 2 Let a be a positive real number.Then we have

Nqeq(at)

(u;v) = 1

vau, au<v. (30)

Proof By aid of theeqanalogue of theq-exponential function, we write

In series representation, (31) can be written as

(6)

By (29), (32) gives a geometric series expansion, and hence we have (Nqeq(at))(u;v) =

1

v

0 (auv) n= 1

vau,au<v.

This finishes the proof of the lemma.

Theorem 3 Let a be a positive real number.Then we have

NqEq(at)

(u;v) =1

v

0

qn(n– 1)

2

au

v n

. (33)

Proof In view of theEqanalogue of the exponential function, we indeed get

Nq(at)

(u;v) =

0

anqn(n2–1)

([n]q)!

1

u

0 tnEq

qv ut

dqt. (34)

The parity of (29) gives (NqEq(at))(u;v) =1v

qn(n2–1)(auv)n.

This completes the proof of the theorem.

The hyperbolic q-cosine and q-sine functions are given as coshqt = eq(t)+eq(–t)

2 and

sinhqt=eq(t)–eq(–t)

2 . Hence, as a corollary of Theorem 2, we have

Nqcoshqt

(u;v) = v

v2+a2u2, au<v, (35)

Nqsinhqt

(u;v) = au

v2+a2u2, au<v. (36)

Theorem 4 Let a be a positive real number.Then we have(Nqcosqat)(u;v) = v2av2u2

pro-vided au<v.

Proof On account of (15) and (29), we obtain

Nqcosqat

(u;v) =

0 a2n

u([2n]q)!

0 t2nEq

qv ut

dqt

=1

v

0

au v

2n

.

Forau<v, the geometric series converges to the sum

Nqcosqat

(u;v) = v

v2a2u2 (37)

providedau<v.

This finishes the proof. Similarly, by (15) and (29), we deduce that

Nqsinqat

(u;v) = au

(7)

5 Nqandq-differentiation

In this section, we discuss someq-differentiation formulae. On account of (12), we derive the following differentiation result.

Lemma 5 Let u,v> 0.Then we have

Proof By using theq-representation ofEqin (12), we write

DqEq

The natural transform of theq-derivativeDqf can be written as follows.

Theorem 6 Let u,v> 0,then we have Proof Using the idea ofq-integration by parts and the formula in (9), we write

Nq

The parity of Lemma 5, gives

(8)

By virtue of (12), (41) yields Nq(Dqf(t))(u;v) = –f(0) + uv

0 f(t)Eq(–q

v

ut)dqt. Hence,

Nq(Dqf(t))(u;v) = –f(0) +vuNq(f)(u;v). This finishes the proof.

Now we extend Theorem 6 to thenth derivative as in the following theorem.

Theorem 7 Let u,v> 0and nZ+.Then we have

Nq

Dnqf(t)(u;v) = v

n

un

Nq(f)

(u,v) –

n–1

i=0

u v

n–1–i

Diqf(0). (42)

Proof On account of Theorem 6, we can writeNq(D2qf(t))(u;v) = –Dqf(0) + uv(–f(0) + v

u(Nqf)(u;v)). Hence we have obtained

Nq

D2qf(t)(u;v) = –Dqf(0) –

v uf(0) +

v2

u2(Nqf)(u;v). (43)

Proceeding as in (43), we obtainNq(Dnqf(t))(u;v) = v 2

u2Nq(f)(u;v) –

n–1

i=0(vu)n–1–iDiqf(0).

This finishes the proof.

6 Nqofq-convolutions

Let functionsf andgbe given in the formf(t) =andg(t) =tβ–1forα,β> 0. We define

theq-convolution off andgas

(fg)q(t) =

t

0

f(τ)g(t)dqt. (44)

Theq-convolution theorem can be read as follows.

Theorem 8 Letα,β> 0.Then we have

Nq

(fg)q

(u;v) =u2Nq

(u,v)N q

–1(u;v). (45)

Proof By aid of (44) and (14), we get Nq((fg)q)(u;v) = Bq( α+1,β)

u

0 t

α+βE

q(–qvtu)dqt.

Hence, by (28) we obtain

Nq

(fg)q

(u;v) =q(α+ 1)q(β)

+β+1

+β+1. (46)

Motivation on (46) givesNq((fg)q)(u;v) =u2(Nqtα)(u,v)(Nqtβ–1)(u;v). The proof is

there-fore completed.

In a similar way, we extend theq-convolution to functions of a power series form.

Theorem 9 Let f(t) = ∞0 aitαi and g(t) = –1. Then we have Nq((fg)q)(u;v) =

u2(N

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Proof Under the hypothesis of the theorem and Theorem 8, we write

Hence, the proof of the theorem is finished.

7 Nqand Heaviside functions

The Heaviside function is defined by

´ whereais a real number.

In this part of the paper, we establish the following theorem.

Theorem 10 Ifu denotes the Heaviside function and u´ ,v> 0,then we have

Nq

Upon integrating and using simple calculation, the above equation reveals

Nq

mation from 0 gives

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8 Theq-analogue of the second kind

Over the setA={f(t)|∃M,τ1,τ2> 0,|f(t)|<Met/τj,t∈(–1)j×[0,∞)}, theq-analogue of

theN-transform of the second type is expressed as

Nqf(u;v) =

0

1

uf(t)eq

v

ut

dqt. (49)

As theq-analogue of the gamma function of the second kind is defined as

γq(t) =

0

xt–1eq(–x)dqx, (50)

it follows that

γq(1) = 1,γq(t+ 1) =qt[t]qγq(t), γq(n) =q

n(n–1)

2 q(n), (51) qbeing theq-analogue of the gamma function of the first kind.

In the remainder of this section, we derive certain results analogous to the ones we have obtained in the previous chapters.

Lemma 11 Letα> –1.Then we have

Nqtα(u;v) = u

α

+1γq(α+ 1). (52) In particular,

Nqtn(u;v) = u

n

vn+1q

n(n–1)

2 [nq]!. (53)

Proof Letα> –1. Then, by the change of variables, we have

Nqtα(u;v) = 1

u

0 tαeq

vt u

dqt=

+1 ∞

0

tαeq(–t)dqt.

By aid of (50), we get (Nqtα)(u;v) =

+1γq(α+ 1). Proof of the second part of the theorem

follows from (51). Hence, we have completed the proof of the theorem.

Theorem 12 Let aR,a> 0,then we have

Nqeq(at)

(u;v) = 1

uv

0 anun

vn qn(n–1)

2 . (54)

Proof By (13) we write

Nqeq(at)

(u;v) = 1

u

0

eq(at)eq

v

ut

dqt=

1

u

0 an

([n]q)!

0 tneq

v

ut

dqt.

By aid of Theorem 2, we get (Nqe

q(at))(u;v) =uv1

0 a

nun

vn q

n(n–1) 2 .

(11)

Theorem 13 Let a> 0,aR,then we have

Proof After simple calculations and using Theorem 2, we obtain

Since the above series determines a geometric series, it reads (Nqe

q(at))(u;v) =uv1 1–1au v =

1

u(vau),au<v.

Hence the theorem is proved.

TheNqtransform ofcosqandsinqis given as follows.

Theorem 14 Let a> 0.Then we have

Proof Using the definition ofcosq, we write

This completes the proof of the theorem.

TheNqtransform ofsinq(at) is given as follows.

(12)

9 Nqofq-differentiation

Before we start investigations, we first assert that

Dqeq

In detail, this can be interpreted as

Dqeq

This proves the above assertion. Hence we prove the following theorem.

Theorem 16 Let u,v> 0.Then we have

Proof By (58), the above equation gives

This completes the proof. Now, we extend (59) to have

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Proceeding to the nth derivative, we get (NqDn

qf)(u;v) = (uv)nqn(Nqf)(qnv;u)

ni=0–1(vu)n–1–iDi qf(0).

This completes the proof of the theorem.

10 Conclusion

Theq-analogues of theN-transform have been applied to various classes of exponential functions, trigonometric functions, hyperbolic functions and certain differential operators as well. As theN-transform defines an extension to Laplace and Sumudu transforms, our results generalize the results existing in the literature.

Acknowledgements

The author would like to express many thanks to the anonymous referees for their corrections and comments on this manuscript.

Funding Not applicable.

Availability of data and materials Not applicable.

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 19 September 2017 Accepted: 22 December 2017

References

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(2011)

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76(A) III, 235-242 (2006)

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