Abstract
In this paper, we discuss the relationship between the Zain Ul Abadin Zafar (ZZ) transform with Laplace and Aboodh transforms. Further, the ZZ transform is applied to the fractional derivative with the Mittag-Leffler kernel defined in both the Caputo and Riemann-Liouville sense. In order to illustrate the validity and applicability of the transform, we solve some illustrative examples.
1. Outline and Motivation
In recent years, fractional calculus (FC) has gained considerable achievements in various fields of science and engineering. Many physical problems [1–7] are modeled by using fractional differential equations (FDE) more accurately than classical differential equations [8–11]. Earlier, various real-life problems were modeled by using the Caputo and Riemann-Liouville (R-L) fractional derivatives. However, Caputo and Fabrizio proposed a new idea that reflects the exponential kernel [12] to address a new way of modeling phenomena with non-local effects. Further, in [13], a new fractional operator (AB) with a Mittag-Leffler kernel was developed. So, in this regard, many researchers [14–16] have given their interest in this definition to solve various problems/models. In fact, in modeling real phenomena, we need a variety of fractional operators to thoroughly describe the complexity of the problem studied. Some other studies regarding fractional calculus and special functions can be found in the literature [17–26].
In the present study, we establish the relationship between the ZZ transform (ZZT) with the Aboodh transform (AT), and the Laplace transform (LT) having their various applications given in [27–31]. Next, the ZZT has been applied to AB fractional operators defined in the Caputo and R-L sense, which are described in terms of theorems. Later, we have solved some test examples defined in the AB sense using this ZZT. The contribution of the present authors to this manuscript are (i) firstly establishing the relationship among ZZT, LT, and AT, (ii) secondly applying ZZT to fractional differential equations defined in the AB derivative to get the solution of the problems. The ZZ transform is the generalization of some famous transforms and we can relate this transformation to other well-known transforms. If we divide the ZZ transform by the transformed variable, then we get the Natural transform. Similarly, relations with other integral transforms in terms of theorems have been included in this paper. The main benefit of this transformation is that it may converge to the Sumudu transform and is advantageous in solving FDEs with variable coefficients.
The organization of the paper is as follows: In section Preliminaries and Basic Definitions, we establish the connection between the Aboodh and ZZ transform; we prove some significant results and create the relationships between AB derivatives with ZZT. In section Applications, some FDEs are solved using ZZT. Finally, a conclusion section is included in section Conclusion.
2. Preliminaries and Basic Definitions
Definition 2.1
The Aboodh transform is obtained on the set of functions
Theorem 2.1
Let us consider G and Fas the Aboodh and Laplace transforms of f (t) ∈ B then [32]
The ZZT was introduced by Zain Ul Abadin Zafar [29, 30]. It generalizes the Aboodh and Laplace integral transforms. In the following definition, we discuss the definition of ZZT.
Definition 2.2 (ZZ Transform)
Suppose f (t) ∀ t ≥ 0 is a function then the ZZT Z (v, s) of f (t) is defined as [29, 30]
Similar to the Aboodh and Laplace transforms, the ZZT is also linear. The MLF is an extension of exponential function which is defined as.
Definition 2.3
Let us consider a function ξ(x, t) ∈ H1(a, b), then for α ∈ (0, 1), the Atangana-Baleanu Caputo (ABC) derivative is written as [13].
Definition 2.4
Let ξ (x, t) ∈ H1(a, b), then for α ∈ (0, 1), the Atangana-Baleanu Riemann-Liouville (ABR) derivative is given as [13]
where ψ (α)is a function with the conditions ψ (0) = ψ (1) = 1and b > a.
Theorem 2.2
The LT of ABC and ABR derivative are, respectively, given as [13]
and
The following theorems have been proposed where it is assumed that f (t) ∈ H1(a, b), b > aand α ∈ (0, 1).
Theorem 2.3
The AT of ABC derivative is given as.
Proof: Using Theorem 2.1 and Equation. (2.2), we may get the desired result.
Theorem 2.4
The Aboodh transform of ABR derivative is written as.
Proof
Applying the Theorem 2.1 and Equation (2.3), we obtain the required result.
The connection between the transforms of Aboodh and ZZ is given in the theorem below.
Theorem 2.5
If G(s) andZ (v, s) are the Aboodh and ZZ transforms of f (t) ∈ B. Then, we obtain
Proof. From the definition of ZZ transform we have
Substituting vt = Ï„ in Equation (2.6) we get
The right-hand side of the above Equation (2.7) may be written as.
where F (.) denotes the Laplace transform of f (t).
Applying the Theorem 2.1, Equation (2.8) can be expressed as
where G(.) denotes the Aboodh transform of f (t).
Theorem 2.6
ZZ transform of f (t) = tα−1 is given as
Proof. The Aboodh transform of f (t) = tα, α ≥ 0 is
Using Equation (2.9), we obtain.
Theorem 2.7
Let α, ω ∈ C and Re (α) > 0, then the ZZ transform of is given as
Proof. We know that Aboodh transform of is written as.
Using the Theorem 2.9, we obtain.
Theorem 2.8
If G(s) and Z (v, s) are the Aboodh and ZZ transforms of f (t). Then the ZZT of ABC derivative is written as.
Proof. Using the Equations (2.1) and (2.4), we have
So, the ZZ transform of ABC is given as.
Theorem 2.9
Let us assume thatG(s) and Z (v, s) are the Aboodh and ZZ transform of f (t). Then the ZZ transform of ABR derivative is given as
Proof. Using the Equations (2.1) and (2.5), we get
From the Equation (2.9), the ZZ transform of ABR is written as.
3. Applications
Let us consider the following initial value problem (IVP) defined in ABC sense [15]
Suppose Z (v, s) and T (v, s) are the ZZ transforms of y (t)and f, respectively. Then by taking the ZZT on both sides of Equation (3.1) and using Equations (2.9) and (2.14), we may get
Then, by applying the inverse ZZT on both sides of Equation (3.2), we obtain the exact solution.
Similarly, we may solve Equation (3.1) defined in ABR derivative.
Example 3.1
Let us take the following fractional IVP [15]
Firstly, we apply the ZZT on both sides of Equation (3.3) which gives
Simplifying Equation (3.4) and using the initial condition, we have
The simplification of Equation (3.5) gives us the following:
Equation (3.6) may be rewritten as.
Applying the inverse of the ZZT on Equation (3.7) and using Equation (2.11), Equation (3.7) is reduced to
where Eα(t) is the MLF.
Substituting α = 1 in Equation (3.8), we obtain
which is the exact solution of Equation (3.3) when α = 1.
Example 3.2
Considering the following fractional IVP [15]
Taking the ZZT on both sides of Equation (3.10) and plugging the initial condition, we get
Applying inverse ZZT on both sides of Equation (3.11), we obtain
It is noticed that if we put α = 0, then Equation (3.12) reduces to y (t) = ηt and substituting α = 1 in Equation (3.12), we obtain . Plugging α = 0.5, we get .
4. Conclusion
In this manuscript, the ZZT is debated and the associated properties of ZZT are established. Some theorems related to the connection between the ZZ, Aboodh, and Laplace transforms are successfully proven. ZZT was applied to FDEs within the AB derivatives. Besides, some fractional initial value problems are solved in order to illustrate the validity and performance of this transformation.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.
Author contributions
RJ: conceptualization, writing–original draft, methodology, software, and validation. SC and DB: project administration and supervision. MA and DB: funding acquisition. SC, DB, and MA: contributed to the analysis, discussion of the results, and help in revision. All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
Acknowledgments
The first author would like to acknowledge the Department of Science and Technology of the Government of India for providing financial support under the scheme of the INSPIRE Fellowship (IF170207) to carry out the present research.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
ZZ transform, fractional calculus, aboodh transform, non-singular kernel, mittag-leffler kernel
Citation
Jena RM, Chakraverty S, Baleanu D and Alqurashi MM (2020) New Aspects of ZZ Transform to Fractional Operators With Mittag-Leffler Kernel. Front. Phys. 8:352. doi: 10.3389/fphy.2020.00352
Received
25 February 2020
Accepted
24 July 2020
Published
23 September 2020
Volume
8 - 2020
Edited by
Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria
Reviewed by
Praveen Agarwal, Anand International College of Engineering, India; Yang Liu, Inner Mongolia University, China; Shilpi Jain, Poornima College of Engineering, India
Updates
Copyright
© 2020 Jena, Chakraverty, Baleanu and Alqurashi.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Snehashish Chakraverty sne_chak@yahoo.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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