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^[1] to solve differential equations and control engineering problems. It is equivalent to the Laplace–Carson transform with the substitution p = 1/u. Sumudu is a Sinhala word, meaning “smooth”.

1 Formal definition
2 Properties and theorems
3 Relationship to other transforms
4 Practical importance
5 See also
6 References

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:

$S\{f(t)\} = F_s(u) = \int_0^\infty (1/u)e^{-t/u}f(t)\,dt.\qquad(1)$

Watugala^[1] first advocated the transform as an alternative to the standard Laplace transform, and gave it the name Sumudu transform. It was early adopted by Weerakoon,^[2] and later by others.^[3]

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 tⁿ 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 > 0 to form the function f(ct) gives a transform F(cu) which is the result of scaling by the same factor.
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

$L\{f(t)\} = F(s) = \int_0^\infty e^{-st}f(t)\,dt\qquad(2)$

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

$C\{f(t)\} = G(p) = \int_0^\infty pe^{-pt}f(t)\,dt.\qquad(3)$

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,^[4] and the Laplace–Carson form remains the standard in Eastern Europe.^[5] 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).^[6]

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

$F_s (u) = \frac{F\left(\frac{1}{u}\right)}{u}$

and its inverse,

$F(s) = \frac{F_s\left( \frac{1}{s} \right)}{s}. \,$

It is thus possible to take a table of Laplace transforms^[4] 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) ^[7]^[8].

It is sometimes said that the Sumudu variant of the Laplace transform is more suitable for educational purposes than is the standard Laplace.^[2]^[1] 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.

Practical importance

In mechanical and material engineering, the Laplace–Carson transform

$C\{f(t)\} = G(p) = \int_0^\infty pe^{-pt}f(t)\,dt\qquad(3)$

is used in the study of the behavior of linear visco-elastic materials. When the linear visco-elastic constitutive law is transformed to the Laplace–Carson domain, its integral form reduces to the simple $\sigma (p)=E (p) \epsilon (p)$ . This is not the case when using the Laplace transform itself. Some other constitutive laws are more appropriately described by the Carson transform,

$Car\{f(t)\} = G_{Car}(p) = p\int_0^\infty e^{-pt}f(t)\,dt\qquad(4)$

with the p in front of the integral.

References

^ ^a ^b ^c Watugala, 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.
^ ^a ^b Weerakoon, S., “Application of Sumudu transform to partial differential equations” International Journal of Mathematical Education in Science and Technology 25 (1994), 277–283.
^ Hussain, 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.
^ ^a ^b Oberhettinger, F. and Badii, L., Tables of Laplace transforms (Berlin: Springer, 1973).
^ Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus (Oxford: Pergamon, 1965).
^ Balser, W., From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582 (Berlin: Springer, 1994), Section 2.1.
^ 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.