Abstract
With the study of extensive literature on the Laplace transform with one and two variables and its properties, applications are available, but there is no work on -dimensional Laplace transform. In this research article, we define -dimensional fractional frequency Laplace transform with shift values. Several theorems are derived with properties of the Laplace transform. The results are numerically analyzed and discussed through MATLAB.
1. Introduction
The 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. This is because of the results reported by these researchers as consequences of their attempts to model real-world phenomena using the fractional operators [1–4]. The 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 [5–17] and the references therein).
The integral transforms such as Mellin, Laplace, and Fourier were applied to obtain the solution of differential equations. These transforms made effectively possible changes of a signal in the time domain into a frequency s-domain in the field of digital signal processing (DSP) [18]. The 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 and obtained the Laplace transform of as . In practice, many applications of Laplace transform (LT), , and the forward discrete Laplace transform (DLT), , are discussed and mentioned by several authors in [20–23]. For physical applications of Laplace transform, refer [24–27].
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 using the generalized difference operator and obtained the outcomes of polynomial and trigonometric functions. In this fractional frequency Laplace transform, the shift values 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 -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 .
2. Preliminaries
Here, we present some basic definitions and results which will be used further.
Definition 1. Let be the function with -variables and be the shift values. Then, the -dimensional partial difference operator is defined aswhere and .
Definition 2. (see [30]). For and , the rising -polynomial factorial function is defined aswhere , is the Euler gamma function, and as the division at a pole yields zero.
Lemma 1. (see [31]). Let and and be real-valued bounded functions. Then,
Theorem 1. (see [31]). Let , , and ; then,
2.1. Notations
(i)(ii)(iii) denotes the set of all subsets of (iv) denotes the number of digits in
Definition 3. (infinite inverse principle law). For the function , we define the infinite inverse principle law as follows:where . In particular, if , we obtain
Theorem 2. Let ; then,Proof. Taking in (1) for the shift value , we obtainIn (8), applying on both sides, we get
Proceeding like this for the induction on , we get the proof of (7).
Corollary 1. Let . Then, we have
Proof. In the proof of Theorem 2, replacing by , we get (10).
Corollary 2. In Theorem 2, applying , we get
Example 1. Let in (11); we get the result for the shift values and asSumming from 0 to for and on both sides yieldsFor , the infinite principle law readsEquating (13) and (14) for the function , we obtainwhich is verified for the particular values , and by MATLAB coding as follows: .
3. -Dimensional Fractional Frequency Laplace Transform
Definition 4. For the function with -variables , the -dimensional fractional frequency Laplace transform is defined as
Remark 1. (i)The -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.
Theorem 3. Let , be a fraction, and . Then, we haveProof. Taking in (16) yields(using Corollary 1), which completes the proof.
Theorem 4. Let , be a fraction, and ; then,Proof. Taking in (16), we havewhich gives (19).
Example 2. For , the summation solution of the exponential function given by the infinite inverse principle law and the closed form of the solution given asis numerically verified for the particular values , and by MATLAB coding as follows: .
The following are the graphical representations of the exponential function in time and frequency domains. Figure 1 is the graphical representation of the input function in the time domain, and Figure 2 is the graphical representation of the output in the frequency domain for the particular values of . One can easily choose the values of fraction to get the output in the frequency domain.
Theorem 5. Let , be a fraction, and ; then,where for , for , and .
Proof. From the previous theorem, we obtain the Laplace transform for the trigonometric function as
The proof then can be continued by making use of the conjugate and the product of each term in -variables.
Theorem 6. Let , be a fraction, and ; then,
Proof. The proof follows by taking in Theorem 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.
Example 3. Equation (24) is the closed-form solution of the sine function. Now, for , the summation solution of the sine function given by the infinite inverse principle law isNow, (24) and (26) are numerically verified for the particular values , and by MATLAB coding as follows: .
The following are the graphical representations of the sine function in time and frequency domains. Figure 3 represents the input time-domain signal (function) for the sine function. Figure 4 represents the output in the frequency domain for the particular values of . Figure 5 represents the output in the frequency domain for the particular values of . Figure 6 represents the output in the frequency domain for the particular values of . Similarly, one can analyze the solution in the frequency domain by choosing diverse values of fraction .
Theorem 7. Let , be a fraction, and ; then,Proof. Taking in (16), we haveby Lemma 1, which gives (4).
3.1. -Kind Riemann Zeta Function in the Discrete Case
In Theorem 7, when and for ., we get
We know that the Riemann zeta function is defined as
Equation (29) can be written as
Taking summation on , on both sides, we getwhich is the product of -kind Riemann zeta function in the discrete case.
3.2. One-Dimensional Laplace Transform on the Fractional Difference Equation
Let and be the two functions. The Leibniz rule of noninteger order is . Here, we present the product formula on the fractional difference operator as .
The following theorem plays an important role in solving the fractional difference equation by one-dimensional Laplace transform.
Theorem 8. Let be a real-valued function and . Then, we have
Proof. Taking in (16), we get . Now, applying (3) and solving, we get
Again taking , using (3) and (16), and applying (34) give
Continuing this process for integer , we arrive at
Since the order is a fraction, we consider (36) for fraction as mentioned in (33).
4. -Dimensional Inverse Laplace Transform
The -dimensional inverse Laplace transform is defined by
Since we can easily represent the -dimensional Laplace transform of the functions mentioned, we can present some results listed as follows:
5. Results and Discussion
When , , and , in all the above results, we have(i).(ii).(iii).(iv).
Similarly, the following result for the hyperbolic functions can be obtained:(i).
These results match the formulas of the double Laplace transform of functions available in the literature.
6. Conclusion
The fractional frequency is used to derive the -dimensional Laplace transform with shift values , 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. The major application of this research work is also provided by considering the classical Laplace transform according to particular values of which are and .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was catalyzed and financially supported by the Tamil Nadu State Council for Science and Technology, Department of Higher Education, Government of Tamil Nadu.