Sumudu transform

In mathematics, the Sumudu transform, is an integral transform similar to the Laplace transform, introduced in the early 1990s by Gamage K. Watugala to solve differential equations and control engineering problems.

Formal Definition

The Sumudu transform of a function "f"("t"), defined for all real numbers "t" ≥ 0, is the function "F"_s("u"), defined by:

: $S\{f(t)\}\; =\; F\_s(u)=\; int\_0^infty\; (1/u)e^\{-t/u\}f(t),dtqquad(1).$

WatugalaWatugala, G. K., “Sumudu transform: a new integral transform to solve differential equations and control engineering problems.” International Journal of Mathematical Education in Science and Technology 24 (1993), 35-43.] first advocated the transform as an alternative to the standard Laplace transform, and gave it the name Sumudu transform (Sumudu is a Sinhala word, meaning “smooth”). It was early adopted by Weerakoon,Weerakoon, S., “Application of Sumudu transform to partial differential equations” International Journal of Mathematical Education in Science and Technology 25 (1994), 277-283.] and later by othersHussain, M. M., and Belgacem, F. M., "Transient solutions of Maxwell's equations based on Sumudu transform," Progress In Electromagnetics Research, PIER 74, 273-289, 2007.] .

Properties and theorems

*The transform of a Heaviside unit step function is a Heaviside unit step function in the transformed domain.
*The transform of a Heaviside unit ramp function is a Heaviside unit ramp function in the transformed domain.
*The transform of a monomial "t"^"n" is the scaled monomial "S"{"tⁿ"} = "n"!·"uⁿ".
*If "f"("t") is a monotonically increasing function, so is "F"("u") and the converse is true for decreasing functions.
*The Sumudu transform can be defined for functions which are discontinuous at the origin. In that case the two branches of the function should be transformed separately. If "f"("t") is "Cⁿ" continuous at the origin, so is the transformation "F"("u").
*The limit of "f"("t") as "t" tends to zero is equal to the limit of "F"("u") as "u" tends to zero provided both limits exist.
* The limit of "f"("t") as "t" tends to infinity is equal to the limit of "F"("u") as "u" tends to infinity provided both limits exist.
* Scaling of the function by a factor "c" to form the function "f"("ct") gives a transform "F"("cu") which is the result of scaling by the same factor.
* As a consequence of the property just described, the Sumudu tranform of an even function is even and a similar rule holds for odd functions.
* By taking the Sumudu transform of the output signal of a dynamic system when the input is a unit step, the transfer function of the dynamic system in the "u"–domain can be defined. This is an easily comprehensible concept for the transfer function of a system.

All of these properties may be deduced from the corresponding properties of the Laplace transform using no more than simple high school algebra.

Relationship to other transforms

The Sumudu transform is a simple variant of the Laplace transform

: $L\{f(t)\}\; =\; F(s)=\; int\_0^infty\; e^\{-st\}f(t),dtqquad(2)$

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

: $C\{f(t)\}\; =\; G(p)=\; int\_0^infty\; pe^\{-pt\}f(t),dtqquad(3)$

The three transforms can be compared by their action on common functions, such as the monomials "t"^"n":
*"L"{"tⁿ"}("s") = "n"!·"s"^−("n"+1)
*"C"{"tⁿ"}("p") = "n"!·"p"^−"n"
*"S"{"tⁿ"}("u") = "n"!·"u"^"n".

Equation (2) is employed in Western countries,Oberhettinger, F. and Badii, L., Tables of Laplace transforms (Berlin: Springer, 1973).] and the Laplace-Carson form remains the standard in Eastern Europe.Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus (Oxford: Pergamon, 1965).] 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).Balser, W., From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582 (Berlin: Springer, 1994), Section 2.1.]

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

$F\_s\; (u)\; =\; F(1/u)/u$ and its inverse,

$F(s)\; =\; F\_s(1/s)/s$.

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) Deakin, M. A. B., “The ‘Sumudu transform’ and the Laplace transform.” International Journal of Mathematical Education in Science and Technology 28 (1997), 159.] [Weerakoon, S., "The 'Sumudu transform' and the Laplace transform - Reply" International Journal of Mathematical Education in Science and Technology Vol 28 Issue 1 (1997), 160.] .

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.

References

ee also

* List of transforms