Sumudu Transform - Relationship To Other Transforms

Relationship To Other Transforms

The Sumudu transform is a simple variant of the Laplace transform

which is also used in its so-called p-multiplied form (sometimes known as the Laplace–Carson transform):

The three transforms can be compared by their action on common functions, such as the monomials tn:

• L{tn}(s) = n!·s−(n+1)
• C{tn}(p) = n!·p−n
• S{tn}(u) = n!·un.

Equation (2) is employed in Western countries, and the Laplace–Carson form remains the standard in Eastern Europe. The Sumudu transform is thus a minor variant of form (3) in which p is replaced by 1/u and in this guise has been pressed into service for special purposes in the form shown in Equation (1).

There are many interconnections between the various transforms. For example, the Mellin transform can by a change of variable be turned into a bilateral version of the Laplace. However, because the ranges of integration differ between the bilateral case and the standard one, the convergence and other properties of the Laplace and the Mellin transforms are also quite different. Similar distinctions apply to other connections between all the usual transforms.

In contrast, the Sumudu transform is essentially identical with the Laplace. Given an initial f(t), its Laplace transform F(s) can be translated into the Sumudu transform F_s(u) of f by means of the relation

and its inverse,

It is thus possible to take a table of Laplace transforms and rewrite it line by line as a table of Sumudu transforms (and vice versa). Similarly, every property proved of the Laplace transform may routinely be turned into a corresponding property of the Sumudu transform (and again vice versa). This proves the essential identity of the two transforms (Sumudu and Laplace).

It is sometimes said that the Sumudu variant of the Laplace transform is more suitable for educational purposes than is the standard Laplace. The argument for this viewpoint proceeds mostly from the somewhat simpler form for the transform of tn and the unit-preserving property of the Sumudu transform. However, even if this were so, the standard versions, Equations (2) and (3), are now so deeply entrenched that change is probably infeasible.