JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi 10.1155/2019/5942139 5942139 Research Article On Katugampola Fourier Transform Salim Tariq O. https://orcid.org/0000-0003-3796-1887 Abu Hany Atta A. K. El-Khatib Mohammed S. Huang Nan-Jing Department of Mathematics Al-Azhar Universiry-Gaza Gaza State of Palestine azhar.edu.eg 2019 1102019 2019 26 06 2019 06 09 2019 1102019 2019 Copyright © 2019 Tariq O. Salim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this article is to introduce a new definition for the Fourier transform. This new definition will be considered as one of the generalizations of the usual (classical) Fourier transform. We employ the new Katugampola derivative to obtain some properties of the Katugampola Fourier transform and find the relation between the Katugampola Fourier transform and the usual Fourier transform. The inversion formula and the convolution theorem for the Katugampola Fourier transform are considered.

1. Introduction

It is well known that fractional calculus is a generalization of the classical integer calculus, where several types of fractional derivatives are introduced and studied such as Riemann–Liouville, Caputo, Hadamard, Weyl, and Grünwald–Letnikov; for more details, one can see [1, 2, 3]. Unfortunately, all these fractional derivatives fail to satisfy some basic properties of the classical integer calculus such as product rule, quotient rule, chain rule, Roll’s theorem, mean value theorem, and composition of two functions. Also, those fractional derivatives inherit nonlocality and most of them propose that the derivative of a constant is not zero. Those inconsistencies lead to some difficulties in the applications of fractional derivatives in physics, engineering, and real-world problems.

To overcome all the difficulties raised, Khalil et al.  introduced and investigated the so-called conformable fractional derivative, and also Katugampola [5, 6] introduced and studied a similar type of derivative, later called the Katugampola derivative and defined as follows.

Definition 1 [<xref ref-type="bibr" rid="B5">5</xref>].

Let ƒ:0, and t>0. Then, the Katugampola derivative of ƒof order α is defined by(1)Dαƒt=limε0ƒteεtαƒtε,where t>0 and α0,1. If f is αdifferentiable in some 0,a,a>0, and limt0+Dαƒt exists, then Dαƒ0=limt0+Dαƒt.

Definition 2 [<xref ref-type="bibr" rid="B5">5</xref>].

Let αn,n+1, for some n, and f be an ndifferentiable at t>0. Then, the αfractional derivative of ƒ is defined by(2)Dαƒt=limε0ƒnt.eε.tnαƒntε,if the limit exists.

Note that the Katugampola derivative satisfies product rule, quotient rule, and chain rule,, and it is consistent in its properties with the classical calculus of integer order. In addition, we have the following theorem.

Theorem 1.

Let αn,n+1, for some n, and ƒ be an n+1differentiable at t>0. Then,(3)Dαƒt=tn+1αƒn+1t.

Proof.

(4) D α ƒ t = lim ε 0 ƒ n t e ε t n α ƒ n t ε = lim ε 0 ƒ n t + ε t n α + 1 + ε 2 t 2 n 2 α + 1 / 2 ! + ε 3 t 3 n 3 α + 1 / 3 ! + ƒ n t ε .

Let h=εtnα+11+εtnα/2!+ε2t2n2α/3!+, then h=εtnα+11+Oε, where h0 as ε0. Hence,(5)Dαƒt=tnα+1limh0ƒnt+hƒnth=tnα+1ƒn+1t.

If αn,n+1, for some n, and ƒ be an n+1differentiable at t>0. Then,(6)Dαƒt=tn+1αƒn+1t.

Note that, for α0,1, t>0, we have Dαƒt=t1αdf/dtt.

The conformable and Katugampola derivatives have been investigated and applied to solve ordinary and partial differential equations of noninteger orders in physics, engineering, and other disciplines; some of these research works have been recently published by Anderson and Ulness , Cenesiz and Kurt , Silva et al. , Yavuz , Yavuz and Yaskiran , Abu Hammad and Khalil , Ilie et al. , and Kurt et al. , and many other valuable works can be found in the literature.

In this research work, we are intended to introduce and study the properties of the Katugampola Fourier transform based on the Katugampola derivative.

2. Katugampola Fourier Transform

From the literature, one can discover that several definitions of fractional Fourier transforms (not necessarily equivalent) have been introduced in recent years. They were motivated by their application to obtain solutions of the problems revealed from quantum mechanics, optics, signal processing, and others.

Negero  had studied applications of Fourier transform to partial differential equations. Also, Çenesiz and Kurt  introduced the definition of conformable Fourier transform. In this section, we define the Katugampola Fourier transform, obtain some properties of this transform, and find the relation between the Katugampola Fourier transform and the usual Fourier transform. We also obtain the formula of the inverse and the convolution theorem for Katugampola Fourier transform.

Definition 3.

Let αn,n+1 and ƒx be a real valued function defined on ,. The Katugampola Fourier transform of ƒx denoted by Fαƒxκ=Fακ,κ, is defined as(7)Fαƒxκ=Fακ=12πeiκxαn/αnƒxxαn1dx.

Theorem 2.

Let αn,n+1 and ƒx be an αdifferentiable real valued function on ,, and ƒx is n+1differentiable at x>0, such that ƒnb=0 as b, and then(8)FαDαƒxκ=iκFαƒnx.

Proof.

By using Definition 3 and integration by parts, we have(9)FαDαƒxκ=Fαxn+1αƒn+1xκ,=12πxαn+1ƒn+1xeiκxαn/αnxnα1dx=12πeiκxαn/αnƒn+1xdx,=12πeiκxαn/αnƒnx+iκ2πeiκxαn/αnƒnxxαn1dx.

But limbƒnb=0, and so(10)FαDαƒxκ=iκ2πeiκxαn/αnƒnxxαn1dx=iκFαƒnx.

The following Lemma is the relation between the Katugampola Fourier transform and the usual Fourier transform.

Lemma 1.

Let αn,n+1 and ƒ:, be a function which satisfies Fαƒxκ=Fακ,κ, property. Then,(11)Fαƒxκ=Fƒαnx1/αnκ,where Fƒxκ denotes the usual Fourier transform defined by(12)Fƒxκ=12πƒxeiκxdx.

Proof.

We have(13)Fαƒxκ=12πƒ(x)eiκxαn/αnxαn1dx.

Then, by making the substitution y=xαn/αn, x=αny1/αn, and dy=xαn1dx, we obtain(14)Fαƒxκ=12πeiκyƒαny1/αndy=Fƒαnx1/αnκ.

Lemma 2.

Let αn,n+1 and Fαƒxκ be the Katugampola Fourier transform of a function f:,. Then, the inversion formula for Katugampola Fourier transform of Fαƒxκ is as follows:(15)Fα1Fαƒxκ=ƒx=12πFαƒxαnαnκeiκxαn/αnƒxdκ.

Proof.

The proof followed by applying the definition of the usual Fourier transform and Lemma 1.

Now, we list down some properties of the Katugampola Fourier transform in the theorem below.

Theorem 3.

Let αn,n+1, ƒ:,, and Fαƒxκ=Mακ, then we have the following:(16)iForα=n+1, wehaveFαƒxaκ=eiaκFƒx,iiFαƒaxκ=1aαnMακaαn,iiiFαeiaxαn/αnƒxκ=Mακa,ivFαeaxαn/αn2κ=12aeκ/4a2,vFαeaxαn/αnκ=2πaa2+κ2,viFαƒxcosaxαnαnκ=12Mακa+Mακ+a,viiFαδxαnαnκ=Fδxκ=12π,where δx is the Dirac delta function.(17)viiiFαƒxκ is a liner operator.

Proof.

The proof is similar to the way as in the usual Fourier transform.

Lemma 3.

Let αm1/m,1,m, and ux,t be an mαdifferentiable real valued function defined on ,. Then,(18)Fαmαxmαux,tκ=iκmFαux,tκ.

Proof.

We can prove this theorem by mathematical induction on m.

For m=1, we have(19)Fααxαux,tκ=iκFαux,tκ,which is true from Theorem 2 with n=0.

Now, assume that the theorem is true for a particular value of m, say r. Then, we have(20)Fαrαxrαux,tκ=iκrFαux,tκ.

Now, we need prove that the theorem is true for r+1; that is,(21)Fαr+1αxr+1αux,tκ=iκr+1Fαux,tκ,and by using Theorem 2 and the assumption, we have(22)Fβr+1αxr+1αux,tκ=Fααxαrαxrαux,tκ,=iκFαrαxrαux,tκ,=iκiκrFαux,tκ,=iκr+1Fαux,tκ.

Therefore, the theorem is true for every positive integer value of m.

Theorem 4 (convolution theorem).

Let gx and hx be arbitrary functions, where ƒ,g:,. Then,(23)Fαghx=FαgxFαhx=FαgFαh,where gh is the convolution of function gx and hx defined as(24)ghx=12πgthxtdt,=12πgxthtdt.

Proof.

By using Lemma 1 and the definition of the Katugampola Fourier transform and changing the order of integration, we obtain the result.

Remark 1.

Let g(x) and hx be arbitrary functions, and let(25)Fα1g^κ=gx,Fα1h^κ=hx,where(26)ƒ^κ=Fαƒxκ=12πeiκxαn/αnƒxxαn1dx.

Then,(27)ghx=Fα1Fαghx=12πgthxtdt,=12πgxthtdt.

3. Katugampola Infinite Fourier Sine and Cosine Transforms

Here we consider the Katugampola infinite Fourier sine and cosine transforms with some of their properties. These transforms are convenient for problems over semi-infinite and some of finite intervals in a spatial variable in which the function or its derivative is prescribed on the boundary.

Definition 4.

(Fourier sine transform). Let αn,n+1 and ƒx be a real valued function. The Katugampola infinite Fourier sine transform of a function ƒ:0,, denoted by Fsαƒx, is defined as(28)Fsαƒxκ=Fsακ=2π0ƒxsinκxαnαnxαn1dx.

Definition 5.

(Fourier cosine transform). Let αn,n+1 and ƒx be a real valued function. The Katugampola infinite Fourier cosine transform of a function ƒ:0,, denoted by Fcαƒx, is defined as (29)Fcαƒxκ=Fcακ=2π0ƒxcosκxαnαnxαn1dx.

Remark 2.

The transforms Fsα and Fcα are liner operators. They are(30)iFsαaƒ+bg=aFsαƒ+bFsαg,iiFcαaƒ+bg=aFcαƒ+bFcαg.

Theorem 5.

Let ƒ:0, be an αdifferentiable real valued function and ndifferentiable at x>0, where ƒnx0 as x and αn,n+1. Then,(31)iFsαDαƒxκ=κFcαƒnx,iiFcαDαƒxκ=κFsαƒnx2πƒn0.

Proof.

i The proof follows by using Definition 4 and Theorem 1 and integration by parts.

ii The proof follows by using Definition 5 and Theorem 1 and integration by parts.

4. Katugampola Finite Fourier Sine and Cosine Transforms

When the physical problem is defined on a finite domain, it is generally not suitable to use transformation with infinite range of integration. In such cases, the usage of finite Fourier transform is very advantageous.

In this section we shall discuss the Katugampola finite Fourier sine and cosine transforms and some of their properties.

Definition 6.

(Katugampola finite Fourier sine transform). Let αn,n+1 for some n and ƒx be a real valued function defined on 0,L. The Katugampola finite Fourier sine transform of ƒx,0<x<L is defined as(32)Fsαƒxκ=Fsακ=0LƒxsinκπxαnLαnxαn1dx.

The inverse Fourier Katugampola sine transform is defined as follows:(33)ƒx=2αnLαnκ=1FsακsinκπxαnLαn.

Definition 7.

(Katugampola Finite Fourier cosine transform). Let αn,n+1 for some n and ƒx be a real valued function defined on 0,L. The Katugampola finite Fourier cosine transform of ƒx,0<x<L is defined as(34)Fcαƒxκ=Fcακ=0LƒxcosκπxαnLαnxαn1dx.

The inverse Fourier Katugampola cosine transform is defined as follows:(35)ƒx=αnLαnFcα0+2αnLαnκ=1FcακcosκπxαnLαn.

Theorem 6.

Letαn,n+1, for some n, and ux,t be a real valued function of two variables 0<x<L and t>0 where ux,t is αdifferentiable and ndifferentiable with respect to x. Then,(36)iFsααuxα=κπαnLαnFcαunx,t,iiFcααuxα=κπαnLαnFsαunx,tun0,tunL,tcosκπ,iiiFsα2αux2α=κπαnLαnFcααunxα,=κπαnLαn2Fsαu2nx,tκπαnLαnu2nL,tcosκπu2n0,t,ivFcα2αux2α=κπαnLαn2Fcαu2nx,tαun0,txααunL,txαcosκπ.

Proof.

i The proof follows by using Definition 6 and Theorem 1 and integration by parts.

ii The proof follows by using Definition 7 and Theorem 1 and integration by parts.

iii , iv By using parts iand ii above, we get the result.

Corollary 1.

Let α0,1 and ux,t be a real valued function of two variables x>0 and t>0. Then,(37)iFsααuxα=κπαLαFcαux,t,iiFcααuxα=κπαLαFsαux,tu0,tuL,tcosκπ,iiiFsα2αux2α=κπαLαFcααuxα,=κπαLα2Fsαux,tκπαLαuL,tcosκπu0,t,ivFcα2αux2α=κπαLα2Fcαux,tαu0,txααuL,txαcosκπ.

Proof.

The proof is direct from Theorem 6 by putting n=0.

Remark 3.

The results we obtained in Corollary 4.4 are similar to the results in (, pp. 137–138).

5. Conclusions

In this paper we obtained several results that have close resemblance to the results found in classical calculus. We defined both the Katugampola infinite and finite Fourier transforms and Fourier sine and cosine transforms. Also we established some properties of these transforms which are considered as generalizations to the usual transform.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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