Extraction of the Laplace, Fourier, and Mellin Transforms from the Sumudu Transform

In this paper, we study connections of the Sumudu transform to other well known transforms, such as the Fourier, Laplace, and Mellin transforms. We afford the Mellin transform a particular attention. Due to its properties, we find that the Sumudu transform may be looked as the source of the other transforms. The sense of this comparison expressed in the paper takes root in the Sumudu -Laplace and Sumudu-Fourier dualities established by Belgacem et al, in the last decade. Our work means that almost in all cases, problems solved any of the three transforms can be equally and perhaps at times more easily treated by the Sumudu transform.