The Intrinsic Structure and Properties of Laplace-Typed Integral Transforms

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 G-transform, the generalized Laplace-typed integral transform, is proposed with the feature of inclusiveness. The proposed Gtransform can provide an adequate transform in a number of engineering problems.


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 Laplacetyped integral transforms. Of course, the structure is dependent on the kernel, and the form of the kernel in Laplacetyped integral transforms is as follows. Laplace transform is defined by Sumudu one is and Elzaki one is Since the Laplace transform m( ) can be rewritten as for = 1/ , we can naturally consider that the form of Laplace-typed integral transform is for an suitable integer : for example, if for > 0. Normally, integral transforms have a base of exponential function and it gets along with how to integrate from 0 to ∞ in order to utilize the property to − or − / converges to 0 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] 2 Mathematical Problems in Engineering are mentioning that Sumudu transform ( = −1) 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 ( = 1) 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 ( = 0) has a strong point in the transforms of derivatives; that is, the differentiation of a function ( ) corresponds to multiplication of its transform m( ) 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 −2. As we checked above, the comprehensive transform in Laplacetyped ones is needed, and thus we would like to proposetransform, 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.

The Properties of Laplace-Typed
Integral Transforms Definition 1. If ( ) is an integrable function defined for all ≥ 0, its generalized integral transform is the integral of ( ) times ⋅ − / from = 0 to ∞. It is a function of , say ( ), and is denoted by ( ); thus

The Definition and the
Let us first check the shifting theorems.
Using ( ) = ⋅ (1/ ) for Laplace transform m( ) = ( ), we can obtain the table of generalized integral transform as shown in Table 1. In the table, we regard Laplacetyped integral transform as a transform. However, we can choose an appropriate constant according to each situation. For example, the choice of = 0 has a merit in the transforms of derivatives, and = −2 has a strong point in the transforms of integrals.
If ( ) is defined and is piecewise continuous on ≥ 0 and satisfies | ( )| ≤ for all ≥ 0, then ( ) exists for all < 1/ . Since the statement is valid.

Transforms of Derivatives and Integrals
Theorem 3. Let a function be -th differentiable. Then the transforms of the first, second, and -th derivatives of ( ) satisfy (1) (2) (3) for is an arbitrary integer.
(4) Let ( ) be piecewise continuous for ≥ 0 and integrable. Then Proof. By the integration by parts, and, similarly, follows.
Continuing this process by substitution as the above and using induction, (3) follows. Let us minutely establish the validity of the statement of (3) by the mathematical induction. For = 1, it clearly follows. Next, we suppose that and we show that ( ( +1) ) can be expressed by In statement (1), the proof of statement Moreover, ( ) clearly satisfies a growth restriction. Solution. Taking -transform on both sides, we have Organizing this equality, we have Simplification gives Since from Table 1, we have the solution for ℎ is hyperbolic functions. From the substitution, above ( ) is exactly a solution of the given equation. Of course, by a simple calculation, the above answer is equal to the solution of [1] which is sin ℎ − .

Convolution and Integral Equations for -Transform.
It is a well-known fact that m( ) ̸ = m( )m( ) and m( )m( ) = m( * ) for * is the convolution of and . This means that convolution has to do with the multiplication of transforms in Laplace transform. Here, we investigate the change at -transform. If two functions and are integrable, the following theorem is held.

Lemma 5 (Lebesgue's dominated convergence theorem (LDCT) [14]). Let ( , , ) be a measure space and suppose that { } is a sequence of extended real-valued measurable functions defined on such that
We note that the above lemma gives validity to the following equality: for ( ) is a nondecreasing sequence.
Proof. Let us put and put Then Let us put = V + , where is at first constant. Then V = − and so we get Since the function is integrable, we can change the order of integration by using Lebesgue's dominated convergence theorem. Hence and when varies to ∞, varies 0 to . Hence It is a well-known fact that convolution helps us to solve integral equations of certain type, mainly Volterra integral equation. Hence, we would like to check the theorem by means of some examples using -transform. Solution. This is rewritten as a convolution: Taking -transform on both sides and applying Theorem 6, we have for = ( ). The solution is and gives the answer by Table 1.

Example 8. Solve the Volterra integral equation of the second kind
Solution. In a way similar to Example 7, the given equation is same as − (1 + ) * = 1 − sinh . Taking -transform, we have hence, Simplification gives and so we obtain the answer by Table 1. is so as well. Similarly, since the solution of is = sinh . Here, we note that the -transform of + 2( * ) = is for = ( ).

Differentiation and Integration of Transforms: ODEs with Variable Coefficients
for = ( ).
(2) Since we have Organizing this equality, we have Since 1/ is the same as , we have for = ( ).
In the above theorem, we note that ( ( ) ) can be represented by (5) From (4), the statement is held for an arbitrary integer .

Heaviside Function
where ℎ is Heaviside function.

Dirac's Delta
If we denote the limit of as ( − ), then
Mathematical Problems in Engineering 7

The Solution of Semi-Infinite String by -Transform
Let us check the solution of semi-infinite string bytransform in terms of a typical example as given in [1].
Example 11 (semi-infinite string). Find the displacement ( , ) of an elastic string subject to the following conditions: (a) The string is initially at rest on -axis from = 0 to ∞.
(b) For > 0, the left end of the string is moved in a given fashion, namely, according to a single sine wave (0, ) = ( ) = sin , if 0 ≤ ≤ 2 and zero otherwise.
Of course there is no infinite string, but our model describes a long string or rope (of negligible weight) with its right end fixed far out on -axis [1].
Solution. It is a well-known fact that the equation of semiinfinite string can be expressed by for / < < / + 2 and zero otherwise, where ℎ is Heaviside function.

Conclusion
This paper has constructed the generalized form of Laplacetyped integral transforms and has established the properties of the generalized Laplace-typed integral transform,transform. The transform is comprehensive form, and it has been well adapted in a number of situations of engineering problems by choosing adequate values in kernel, and we newly presented the value = −2 which is suitable for transforms of integrals. And the future work is to find the other values of which are suitable for each situation. The strong point of this article is in the high applicability to engineering problems.

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.