Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1762729, 8 pages

https://doi.org/10.1155/2017/1762729

## The Intrinsic Structure and Properties of Laplace-Typed Integral Transforms

Faculty of General Education, Kyungdong University, Yangju, Gyeonggi 11458, Republic of Korea

Correspondence should be addressed to Hwajoon Kim

Received 14 February 2017; Revised 12 April 2017; Accepted 10 May 2017; Published 8 June 2017

Academic Editor: Salvatore Strano

Copyright © 2017 Hwajoon Kim. 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.

#### Abstract

We would like to establish the intrinsic structure and properties of Laplace-typed integral transforms. The methodology of this article is done by a consideration with respect to the common structure of kernels of Laplace-typed integral transform, and -transform, the generalized Laplace-typed integral transform, is proposed with the feature of inclusiveness. The proposed -transform can provide an adequate transform in a number of engineering problems.

#### 1. Introduction

The key motivation for pursuing theories for integral transforms is that it gives a simple tool which is represented by an algebraic problem in the process of solving differential equations [1]. In the most theories for integral transforms, the kernel is doing the important role which transforms one space to the other space in order to solve the solution. The main reason to transform is because it is not easy to solve the equation in the given space, or it is easy to find a characteristic for the special purpose. For example, in computed tomography (CT) or magnetic resonance imaging (MRI), we obtain the projection data by integral transform and produce the image with the inverse transform. This is the strong point of integral transform.

To begin with, let us see the intrinsic structure of Laplace-typed integral transforms. Of course, the structure is dependent on the kernel, and the form of the kernel in Laplace-typed integral transforms is as follows. Laplace transform is defined by Sumudu one is and Elzaki one is Since the Laplace transform can be rewritten as for , we can naturally consider that the form of Laplace-typed integral transform is for an suitable integer : for example, if for .

Normally, integral transforms have a base of exponential function and it gets along with how to integrate from to in order to utilize the property to or converges to when approaches . Sumudu/Elzaki transform is a kind of modified Laplace one introduced by Watugala [2] in 1993/Elzaki et al. [3] in 2012 to solve initial value problems in engineering problems [4]. Belgacem et al. [5, 6] are mentioning that Sumudu transform () has scale and unit-preserving properties, and it may be used to solve problems without resorting to a new frequency domain. Elzaki et al. [3, 7, 8] insists that Elzaki transform () should be easily applied to the initial value problems with less computational work and solve the various examples which are not solved by the Laplace or the Sumudu transform. As an application, Agwa et al. [9] deal with Sumudu transform on time scales and its applications, Eltayeb and Kilicman [10] have checked some applications of Sumudu one, and Eltayeb et al. are highlighting the importance of fractional operators of integral transform and their applications in [11]. The shifted data problems, shifting theorems, and the forms of solutions of ODEs with variable coefficients can be found in [4, 12, 13].

On the other hand, Kreyszig [1] says that Laplace transform () has a strong point in the transforms of derivatives; that is, the differentiation of a function corresponds to multiplication of its transform by . In the other view, if we want the inverse case, the transform giving a simple tool for transforms of integrals, then we can choose a suitable form of integral transform such as This means that the integer is applicable to . As we checked above, the comprehensive transform in Laplace-typed ones is needed, and thus we would like to propose -transform, a generalized Laplace-typed integral transform, which is more comprehensive and intrinsic than the existing transforms.

This intrinsic structure in Laplace-typed integral transforms has a meaning which can be directly applied to any situation by choosing an appropriate integer . The main objective of this paper is to construct the generalized form of Laplace-typed integral transforms and establish the properties of it, and, to the author’s knowledge, the proposed -transform is the first attempt to generalize Laplace-typed integral transforms. Finally, we would like to mention that Laplace transform gave many considerations to this article.

#### 2. The Properties of Laplace-Typed Integral Transforms

##### 2.1. The Definition and the Table of Generalized Integral Transform , Shifting Theorems

As mentioned before, let us rewrite the definition of Laplace-typed integral transforms, and we would like to call it -transform.

*Definition 1. *If is an integrable function defined for all , its generalized integral transform is the integral of times from to . It is a function of , say , and is denoted by ; thus

Let us first check the shifting theorems.

Theorem 2. *(1) (-shifting) If has the transform , then has the transform That is, **(2) (-shifting) If has the transform , then the shifted function has the transform . In formula, for is Heaviside function (we write since we need to denote -space). Additionally, holds.*

*Proof. * From we obtain the result.

(2)Substituting , we obtain for is Heaviside function.

By the similar way, we have for is Heaviside function.

Using for Laplace transform , we can obtain the table of generalized integral transform as shown in Table 1. In the table, we regard Laplace-typed integral transform as a transform. However, we can choose an appropriate constant according to each situation. For example, the choice of has a merit in the transforms of derivatives, and has a strong point in the transforms of integrals.

If is defined and is piecewise continuous on and satisfies for all , then exists for all . Since the statement is valid.