n-Dimensional Fractional Frequency Laplace Transform by the Inverse Difference Operator

Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur District 635601, Vellore, India Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Department of Mathematics, Cankaya University, Etimesgut 06790, Ankara, Turkey


Introduction
e fractional calculus is a branch of mathematics that focuses on arbitrary order integrals and derivatives. In spite of that, this type of calculus is as older as the conventional calculus, and it has attracted the interest of researchers for the last few decades. is is because of the results reported by these researchers as consequences of their attempts to model real-world phenomena using the fractional operators [1][2][3][4]. e discrete version of these operators fetched the attention of research studies as well. Many good results were reported when fractional sums and differences were used in studying related problems (see [5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein).
e integral transforms such as Mellin, Laplace, and Fourier were applied to obtain the solution of differential equations.
ese transforms made effectively possible changes of a signal in the time domain into a frequency sdomain in the field of digital signal processing (DSP) [18]. e delta Laplace transform was first defined in a very general way by Bohner and Peterson [19]. In 2015, Ivic discussed the discrete Laplace transforms in the view of fast decay factor e − sx and obtained the Laplace transform of P(x) as ∞ 0 P(x)e − sx dx � πs − 2 ∞ n�1 r(n)e − (π 2 /n) . In practice, many applications of Laplace transform (LT), L[f(x)] � ∞ 0 f(x)e − sx dx, and the forward discrete Laplace transform (DLT), L[f(n)] � ∞ n�0 f(n)e − sn , are discussed and mentioned by several authors in [20][21][22][23]. For physical applications of Laplace transform, refer [24][25][26][27].
In the existing Laplace transform, the shifting value of time domains is one. In 2016, Britto Antony Xavier et al. [28] defined the Laplace transform with shift value ℓ using the generalized difference operator and obtained the outcomes of polynomial and trigonometric functions. In this fractional frequency Laplace transform, the shift values ] j ′ s, j � 1, 2, . . . , n lie in the interval [0, 1]. In [29], the author introduced the double Laplace transform and applied to solve initial and boundary value problems.
In this research work, we extend the work of Laplace transform into an n-dimensional space in discrete case. We present several properties of the fractional transforms for functions such as polynomial factorial, exponential, and trigonometric functions. Also, we derive the relation between Laplace transform and Riemann zeta functions. Furthermore, we present the inverse Laplace transform to compare the results with the existing classical Laplace transform for the particular value of n.

Preliminaries
Here, we present some basic definitions and results which will be used further. Definition 1. Let u(t) be the function with n-variables and h ∈ R n be the shift values. en, the n-dimensional partial difference operator is defined as where t � (t 1 , t 2 , . . . , t n ) and h � (h 1 , h 2 , . . . , h n ).
Definition 2 (see [30]). For h > 0 and μ ∈ R, the rising h-polynomial factorial function is defined as where { } as the division at a pole yields zero.

Corollary 2.
In eorem 2, applying a � 3, we get Example 1. Let n � 2 in (11); we get the result for the shift values h 1 and h 2 as Summing from 0 to ∞ for t 1 and t 2 on both sides yields For n � 2, the infinite principle law reads Equating (13) and (14) for the function which is verified for the particular values coding as follows:

n-Dimensional Fractional Frequency Laplace Transform
Definition 4. For the function u(t) with n-variables t 1 , t 2 , . . . , t n , the n-dimensional fractional frequency Laplace transform is defined as Remark 1 (i) e n-dimensional fractional frequency Laplace transform satisfies the linear property. (ii) In the aforesaid equation (16), we represent the Laplace transform of the functions in two ways: one in the closed-form solution and another one in the summation form solution. In this paper, we numerically verified and analyzed with MATLAB that both solutions are equal.

Example 2.
For n � 2, the summation solution of the exponential function given by the infinite inverse principle law and the closed form of the solution given as e proof then can be continued by making use of the conjugate and the product of each term in n-variables. Theorem 6. Let t ∈ R n , h > 0, ] j be a fraction, and s j > 0, j � 1, 2, . . . , n; then, Proof. e proof follows by taking n � 2 in eorem 5, making the product by its conjugate terms and separating the real and imaginary parts to get the double Laplace transform for the sine and cosine terms. (24) is the closed-form solution of the sine function. Now, for n � 2, the summation solution of the sine function given by the infinite inverse principle law is

n-Kind Riemann Zeta Function in the Discrete Case.
In eorem 7, when ] j � 1 and h j ⟶ 0 for j � 1, 2, . . . , n., we get We know that the Riemann zeta function is defined as Equation (29) can be written as

Mathematical Problems in Engineering
Taking summation on s j , j � 1, 2, . . . , ∞, on both sides, we get which is the product of n th -kind Riemann zeta function in the discrete case.

One-Dimensional Laplace Transform on the Fractional Difference Equation.
Let u(t) and v(t) be the two functions. e Leibniz rule of noninteger order is Here, we present the product formula on the fractional difference e following theorem plays an important role in solving the fractional difference equation by one-dimensional Laplace transform. (16), we get

Theorem 8. Let u(t 1 ) be a real-valued function and
. Now, applying (3) and solving, we get Again taking u(t 1 ) � Δ 2 h 1 u(t 1 ), using (3) and (16), and applying (34) give Continuing this process for integer n, we arrive at Since the order is a fraction, we consider (36) for fraction ] as mentioned in (33).

n-Dimensional Inverse Laplace Transform
e n-dimensional inverse Laplace transform is defined by Since we can easily represent the n-dimensional Laplace transform of the functions mentioned, we can present some results listed as follows:

Conclusion
e fractional frequency is used to derive the n-dimensional Laplace transform with shift values h j , j � 1, 2, . . . , n, that presents more accuracy outputs of the input functions such as exponential, polynomial factorial, polynomial, and trigonometric functions. Also, the numerical results and the solutions are analyzed graphically by MATLAB. e major application of this research work is also provided by considering the classical Laplace transform according to particular values of n which are ] j � 1 and h j ⟶ 0, j � 1, 2, . . . , n.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.