Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 217656, 7 pages

http://dx.doi.org/10.1155/2013/217656

Research Article

On Multiple Convolutions and Time Scales

Hassan Eltayeb,¹ Adem Kılıçman,² and Brian Fisher³

¹Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

²Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

³Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK

Received 23 September 2013; Accepted 7 November 2013

Academic Editor: Suares Clovis Oukouomi Noutchie

Copyright © 2013 Hassan Eltayeb et al. 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.

The properties of the multiple Laplace transform and convolutions on a time scale are studied. Further, some related results are also obtained by utilizing the double Laplace transform. We also provide an example in order to illustrate the main result.

1. Introduction

A time scale is a nonempty closed subset of the real numbers and it has the topology that it inherits from the real numbers with the standard topology. It is also well known that if the time scale is the set of real numbers, then the dynamic equation is a differential equation, whilst if the time scale is the set of integers, then the dynamic equation is reduced to a difference equation. Thus integral transform of differential and integral calculus on time scales allows us to develop a theory of differential equations. For the single Laplace transform on time scales, see [1–3]. Similarly, in [4–8], the authors defined the time scale to be any nonempty closed subset of the real numbers and provided the motivation and formulation of delta derivatives on a time scale as well as the properties of delta derivatives and integrals. In [2], Bohner and Peterson defined the Laplace transform of a time scale for function of single variable as follows.

Definition 1. For , the time scale or generalized Laplace transform of , denoted by or , is given by

where consists of all for which the improper integral exists and for all .

Definition 2. The function is said to be of exponential type I if there exist constants such that . Furthermore, is said to be of exponential type II if there exist constants ,   such that .

The following two theorems were proved in [3].

Theorem 3. The integral converges absolutely for if is of exponential type II with exponential constant . For more details, see [3].

Theorem 4. Let be regressive. Then

(I)

provided that ,

(II)

provided that ,

(III)

provided that .

The following theorem was also studied by Bohner and Peterson in [2].

Theorem 5. If is such that is regulated, then

for all such that .

Throughout the study, the following notation will be useful, where are time scales and

In [2], further it was proved that

was satisfied, where , by using the property

Later in [9], Bohner and Guseinov defined the single convolution operation on time scale of two functions by the following formula:

In this study, we introduce double Laplace transform on time scales and study some of the properties in solving partial differential equations. Further, we also extend the -Laplace transform, which is given by Jackson in [10], to the multiple -Laplace transform as follows.

Definition 6. Let ,  , and . The -double Laplace transform of the function (for ) with respect to is given by

provided that the integral exists, further , and , for all .

Lemma 7. If are regressive, then

Proof. By using the above analysis, we have

As an example, we can easily compute , where are constants such that .

We have

For and with and  , we can easily see that

and then the double Laplace transforms are given by

Similarly, we have

Proposition 8. Suppose ,   are two time scales such that and for some and all . Let and

Then

Proof. Since

and on using the following relation:

we have

Since and , we get

Thus we obtain that

and it follows that

Theorem 9. Let be regulated and let

for . Then

for .

Proof. By using the definition of the double Laplace transform on a time scale, we have

and on using (21), we have

Now, applying integration by parts, and on using the fact that together with the fundamental theorem of calculus, we obtain that

Example 10. Let , , , and for and , and whilst for , and ,  , where and the symbol denotes the tensor product. Then we have

Now assume that such that is continuous. Then the -double Laplace transform

holds for those regressive with respect to and which further satisfies

We next extend the result that was proved by Davis et al. in [3].

Proposition 11. Let , . Assume that is one of the following functions:

If are regressive and satisfy

then

Proof. Let us study the case . By using (21), we get

and by using Theorem 4 and (21) the right hand side of equation (37) is

Then

Theorem 12. Let and suppose that and are continuous for all and , respectively; then the -double Laplace transforms for and are given by

for all with

respectively.

Proof. Let . On using the definition of the -single Laplace transform and integrating parts of the function , we have

Then the single Laplace transform with respect to is given by

Similarly, Laplace transform for (44) with respect to is given by

Then (45) is called the -double Laplace transform for the function . Similarly, if in (41), the -double Laplace transform for the function is given by

Now, in order to obtain the -double Laplace transform for the function

with respect to , we proceed as follows: first of all by taking the -single Laplace transform with respect to the variable and on using the fact that

we have

Thus on using (44), we obtain that

In a similar way, the single Laplace transform with respect to is given by

Further, the -double Laplace transform for the function with respect to is given by

Finally, note that we can generalize the proof for .

Next, we define the double convolution of two time scale functions as follows.

Definition 13. The double convolution of the function ,   is given by

In the next theorem we prove the properties of the double convolution on a time scale as follows.

Theorem 14. Let , be integrable functions on a time scale. Then the following properties are satisfied:

Proof. We can easily show that convolution is commutative by using the definition of the double convolution

For the proof of the associative law, we also use the definition as follows:

Theorem 15 (convolution theorem). Let ,   be regulated functions. Then the double Laplace transform of the double convolution is given by

Proof. By using the definition of the double Laplace transform on a time scale, we obtain that

where and the symbol denotes the tensor product. Thus we have

Finally, we give the solution to the wave equation in one dimension as follows. Consider the wave equation in the form of

under the conditions

Then by formally taking the double Laplace transform, we get

By using partial fractions, we have

Thus the solution to the above equation is given by

In particular, if we consider the case , , for all , then for any constant and so the solution is given by

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of paper.

Acknowledgment

The authors are grateful to the referees for the useful comments regarding presentation in the early version of the paper. The authors also gratefully acknowledge that this project was supported by King Saud University, Deanship of Scientific Research College of Science Research Center.

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