International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2006 / Article

Open Access

Volume 2006 |Article ID 091083 | https://doi.org/10.1155/JAMSA/2006/91083

Fethi Bin Muhammed Belgacem, Ahmed Abdullatif Karaballi, "Sumudu transform fundamental properties investigations and applications", International Journal of Stochastic Analysis, vol. 2006, Article ID 091083, 23 pages, 2006. https://doi.org/10.1155/JAMSA/2006/91083

Sumudu transform fundamental properties investigations and applications

Received03 May 2005
Revised20 Oct 2005
Accepted20 Oct 2005
Published22 May 2006

Abstract

The Sumudu transform, whose fundamental properties are presented in this paper, is still not widely known, nor used. Having scale and unit-preserving properties, the Sumudu transform may be used to solve problems without resorting to a new frequency domain. In 2003, Belgacem et al have shown it to be the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving. Here, using the Laplace-Sumudu duality (LSD), we avail the reader with a complex formulation for the inverse Sumudu transform. Furthermore, we generalize all existing Sumudu differentiation, integration, and convolution theorems in the existing literature. We also generalize all existing Sumudu shifting theorems, and introduce new results and recurrence results, in this regard. Moreover, we use the Sumudu shift theorems to introduce a paradigm shift into the thinking of transform usage, with respect to solving differential equations, that may be unique to this transform due to its unit-preserving properties. Finally, we provide a large and more comprehensive list of Sumudu transforms of functions than is available in the literature.

References

  1. M. A. Asiru, “Sumudu transform and the solution of integral equations of convolution type,” International Journal of Mathematical Education in Science and Technology, vol. 32, no. 6, pp. 906–910, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  2. M. A. Asiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. M. A. Asiru, “Classroom note: application of the Sumudu transform to discrete dynamic systems,” International Journal of Mathematical Education in Science and Technology, vol. 34, no. 6, pp. 944–949, 2003. View at: Publisher Site | Google Scholar
  4. F. B. M. Belgacem, “A generalized Stirling inversion formula,” Algebras, Groups and Geometries, vol. 18, no. 1, pp. 101–114, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications to integral production equations,” Mathematical Problems in Engineering, vol. 2003, no. 3, pp. 103–118, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. L. Debnath, Integral Transforms and Their Applications, CRC Press, Florida, 1995. View at: MathSciNet
  7. R. Merris, “The p-Stirling numbers,” Turkish Journal of Mathematics, vol. 24, no. 4, pp. 379–399, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. A. D. Poularikas, Ed., The Transforms and Applications Handbook, The Electrical Engineering Handbook Series, CRC Press, Florida, 1996. View at: Zentralblatt MATH | MathSciNet
  9. M. R. Spiegel, Theory and Problems of Laplace Transforms, Schaums Outline Series, McGraw-Hill, New York, 1965.
  10. G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35–43, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  11. G. K. Watugala, “Sumudu transform—a new integral transform to solve differential equations and control engineering problems,” Mathematical Engineering in Industry, vol. 6, no. 4, pp. 319–329, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. G. K. Watugala, “The Sumudu transform for functions of two variables,” Mathematical Engineering in Industry, vol. 8, no. 4, pp. 293–302, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. S. Weerakoon, “Application of Sumudu transform to partial differential equations,” International Journal of Mathematical Education in Science and Technology, vol. 25, no. 2, pp. 277–283, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. S. Weerakoon, “Complex inversion formula for Sumudu transform,” International Journal of Mathematical Education in Science and Technology, vol. 29, no. 4, pp. 618–621, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2006 Fethi Bin Muhammed Belgacem and Ahmed Abdullatif Karaballi. 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.

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