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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 480528, 8 pages
http://dx.doi.org/10.1155/2011/480528
Research Article

On the Solution of Distributional Abel Integral Equation by Distributional Sumudu Transform

Department of Mathematics, Faculty of Mathematics, Jai Narain Vyas University, Jodhpur 342 005, India

Received 20 October 2010; Accepted 8 January 2011

Academic Editor: Christian Corda

Copyright © 2011 Deshna Loonker and P. K. Banerji. 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

Solution of the Abel integral equation is obtained using the Sumudu transform and further, distributional Sumudu transform, and, distributional Abel equation are established.

1. Introduction

This section deals with the definition, terminologies, and properties of the Sumudu transform and the Abel integral equation. In Section 2, solution of Abel integral equation is obtained by the application of the Sumudu transform, and in Section 3, the Sumudu transform is proved for distribution spaces, and the solution of Abel integral equation in the sense of distribution is obtained.

The Sumudu transform is introduced by Watugala [1, 2] to solve certain engineering problems. Complex inversion formula for the Sumudu transform is given by Weerakoon [3]. For more of its applications, see [46].

Let be a constant and and need not exist simultaneously (each may be infinite). Then, the set is defined by which initiates the definition of the Sumudu transform, see [4], in the form

In other words, the Sumudu transform can also be written as [4, 6] inversion formula of which is given by

The discrete analog of the Sumudu transform (1.2) for the power series function , having an interval of convergence containing , is given by

The Laplace-Sumudu duality is expressed as where is the Sumudu transform and is the Laplace transform.

The Sumudu transform of th order derivative of is defined by

Convolution of the Sumudu transform is

denotes the Sumudu transform of th antiderivative of , which is obtained by integrating the function    times successively, that is

If is the Sumudu transform of a function in , is the th derivative of with respect to , and is the th derivative of with respect to , then the Sumudu transform of the function is

Some properties of the Sumudu transform, see [6], relevant to present paper may be considered as below.

Lemma 1.1. Let and be continuous functions defined for , possessing Sumudu transforms and , respectively. If almost everywhere, then , where is a complex number.

Theorem 1.2 (existence of Sumudu transform). If is of exponential order, then, indeed, its Sumudu transform exists, which is given by where . The defining integral for exists at point in the right hand plane and .

Proposition 1.3 (Sumudu transform of higher derivatives). Let be times differentiable on and for . Further, suppose that . Then for , , and for any polynomial of degree for . In particular, where by one means a column vector and dom will mean domain.

The Abel integral equation is given by [7, page 43] solution of which is given by The solution can be obtained by two methods which are shown in [7, pages 44-45].

2. Solution of Abel Integral Equation Using Sumudu Transform

In this section, we prove the Abel integral equation using the Sumudu transform. We write the Abel integral equation in the form which can also be written as where , is Heavside's unit step function. Applying the Sumudu transform on both sides of (2.2) and using the convolution of the Sumudu transform (1.8) in (2.2), we have When , the Sumudu transform is . Similarly, we prove that if , then the Sumudu transform is , where , . Putting value of in (2.3), we have That is, That is, where .

By virtue of (1.7), . Invoking it in (2.6), we have that is, that is, This is the required solution of (2.1), the Abel integral equation, by virtue of the Sumudu transform.

3. Sumudu Transform and Abel Integral Equation on Distribution Spaces

This section deals with the Sumudu transform on certain distribution spaces, and, subsequently a relation is established to solve the Abel integral equation by the distributional Sumudu transform.

If is a locally integrable function, then the distribution through the convergent integral ( in ) is defined by

By virtue of Proposition 1.3, the Sumudu transform of the function generates a distribution, or in other words, is in and belongs to , where and denote, respectively, testing function space and its dual.

The linearity property, defined in [4, page 117], is where and are any numbers.

By virtue of (3.1) and (3.2), we state, if the locally integrable functions and are absolutely integrable over and if their Sumudu transforms and are equal everywhere, then almost everywhere.

In what follows is the proof of the Parseval equation for the distributional Sumudu transformation, which will be employed in the analysis of the problem of this paper.

Theorem 3.1. If the locally integrable functions and are absolutely integrable over , then

Proof. Since the transforms and are bounded and continuous for all , as shown in Section 1, therefore both the sides of (3.3) converge. Moreover,
Since the above integral is absolutely integrable, therefore
Further, we consider such that that is,
Thus, the Parseval relation of the Sumudu transform can be written as
This proves the theorem.

It may not be out of place to mention that (1.14) can be attained by virtue of the convolution of distribution for explicit interpretation, refer to [7, page 180]. Consider convolution as a bilinear operation . If and are locally integrable functions, then we have

When and , we have . Thus, the convolution with defines an operator of the space , which is given by where and are locally integrable functions.

The convolution form of (1.14) is where and is locally integrable, since .

Equation (3.11) asserts that the Abel integral equation can be interpreted in the sense of distributions and the functions and can be considered to be elements of . Similarly, (1.15) can also be interpreted in the sense of distribution, given by

It may be remarked that the Sumudu transform has an affinity for the mixed spaces, by virtue of which it is identified, owing to the fact that , and similarly others mentioned above, is one of the mixed distribution space that is identified with the space of distribution , support of which is contained in .

Whereas (3.11) and (3.12) express the solution of Abel integral equation on certain distribution spaces, the similar method is invoked (as in Section 2) to obtain the solution of the Abel integral equation by using the distributional Sumudu transform. The analysis is, therefore, explicitly explained and justified.

Acknowledgments

This paper is partially supported by the DST (SERC), Government of India, Fast Track Scheme Proposal for Young Scientist, no. SR/FTP/MS-22/2007, sanctioned to the first author (D. Loonker) and the Emeritus Fellowship, UGC (India), no. F.6-6/2003 (SA-II), sanctioned to the second author (P. K. Banerji). The authors are thankful to the concerned referee for fruitful comments that improved their understanding.

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