Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 91083, 23 pages

Sumudu transform fundamental properties investigations and applications

Fethi Bin Muhammed Belgacem1 and Ahmed Abdullatif Karaballi2

1Faculty of Information Technology, Arab Open University, P.O. Box 3322, Safat 13033, Kuwait
2Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Received 3 May 2005; Revised 20 October 2005; Accepted 20 October 2005

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.


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.