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.
2.2. 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 and integrable. Then holds for .
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 , it clearly follows. Next, we suppose that and we show that can be expressed by In statement , the proof of statement follows by applying to .
Finally, follows as below.
Show that holds for . Using Theorem 3 and putting , Moreover, clearly satisfies a growth restriction.
Let us check an example in [1] by the -transform.
Example 4. Solve , , and .
Solution. Taking -transform on both sides, we have Organizing this equality, we have Simplification gives Since from Table 1, we have the solution for 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 .
2.3. Convolution and Integral Equations for -Transform
It is a well-known fact that and 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(a) exists -a.e.(b)there is an integrable function so that, for each , -a.e. Then is integrable and
We note that the above lemma gives validity to the following equality: for is a nondecreasing sequence.
Theorem 6.
Proof. Let us put and put Then Let us put , where is at first constant. Then 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 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.
Example 7. Solve the Volterra integral equation of the second kind
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 . Taking -transform, we have hence, Simplification gives and so we obtain the answer by Table 1.
Example 9. Find the solution of
Solution. Since this equation is , taking -transform on both sides, we have for . Thus and so we obtain the solution .
Let us check this by the direct calculation. Expanding the given equation, we have Differentiating both sides twice with respect to , we have . Thus, from and obtained by calculating course, we have the solution .
Similarly, we can easily obtain the solution of integral equations by using . For example, let us consider By the -transform, we have and so, we have the solution for . Of course, this result is the same as the result by using convolution, and the result of the direct calculation is so as well. Similarly, since the solution of is . Here, we note that the -transform of is for .
3. Differentiation and Integration of Transforms: ODEs with Variable Coefficients
Theorem 10. Let us put . Then (1),(2),(3),(4), ,(5).
Proof. (1) Since for .
(2) Since we have Organizing this equality, we have (3)thanks to Lebesgue’s dominated convergence theorem.
(4) In the definition of let us put . Then Since 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 .
4. -Transforms of Heaviside Function and Dirac’s Delta Function: Shifted Data Problems
4.1. Heaviside Function
where is Heaviside function.
4.2. Dirac’s Delta Function
We consider the function if and otherwise. Taking -transform, we have If we denote the limit of as , then
4.3. Shifted Data Problems
For given differential equations , , and , where and and are constant, we can set . Then gives and so we have for input . Taking the transform, we can obtain the output .
5. The Solution of Semi-Infinite String by -Transform
Let us check the solution of semi-infinite string by -transform 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 to .(b)For , the left end of the string is moved in a given fashion, namely, according to a single sine wave , if and zero otherwise.(c)Furthermore, as for .
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 semi-infinite string can be expressed by subject to , as , , and . Taking -transform with respect to and by Theorem 3, we have for . On the other hand, Thus, the equation becomes Since this equation may be regarded as an ODE for considered as a function of , its general solution can be represented by From the initial conditions, we have . In [15, 16], we have dealt with the validity on exchangeability of integral and limit in the solving process of PDEs by using dominated convergence theorem. Hence, we have implying that and so . Thus, By the -shifting theorem (Theorem 2), we obtain the inverse transform for and zero otherwise, where is Heaviside function.
6. Conclusion
This paper has constructed the generalized form of Laplace-typed 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 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.
Acknowledgments
This research was supported by Kyungdong University Research Grant.