Mathematics Department: AN INTRODUCTION OF SUMUDU TRANSFORM

AN INTRODUCTION OF SUMUDU

TRANSFORM

S. Jeevitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Komala

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Silambarasi

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

S. Susitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

R.Vanitha

PG and Research Department of Mathematics

Marudhar Kesari Jain College For Women, Vaniyambadi, Tamil Nadu, India.

ABSTRACT

In this paper, we see the definition, some basic properties and fundamental properties

of Sumudu transform, relationships between Laplace and Sumudu transforms and Existence

of Sumudu transform.

KEYWORDS

Sumudu Transform, Gamma Function, Laplace Transform.

INTRODUCTION

The Sumudu transform is introduced by Watugula. Sumudu transform may be used

to solve problems without resorting to a new frequency domain .Due to its simple

formulation and consequent special and useful properties, the Sumudu transform has

already shown much promise. It is revealed here in and elsewhere that it can help to solve

intricate problems in engineering mathematics and applied sciences. However, despite the

potential presented by this new operator, only few theoretical investigations have appeared

in the literature, over a fifteen-year period. Most of the available transform theory books, if

not all, do not refer to the Sumudu transform. Even in relatively recent well known

comprehensive handbooks, such as Debnath and Poularikas, no mention of the Sumudu

transform can be found.

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SUMUDU TRANSFORM

Watugala introduced a new transform and named as Sumudu transform which is

defined by the following formula

𝐹(𝑢) = 𝒮[𝑓(𝑡); 𝑢] =

𝑢

∫ 𝑒−(

𝑢

) 𝑓(𝑡)𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2)

∞

BASIC SUMUDU TRANSFORM PROPERTIES

Sumudu transform for 𝑓 ∈ 𝐴:

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

Duality with Laplace transforms:

𝐺(𝑢) =

𝐹(1

⁄𝑢)

𝑢

, 𝐹(𝑠) =

𝐺(1

⁄𝑠)

𝑠

Linearity Property:

𝒮[𝑎𝑓(𝑡) + 𝑏𝑔(𝑡) = 𝑎𝒮[𝑓(𝑡)] + 𝑏𝒮[𝑔(𝑡)]]

Sumudu Transform of Function Derivatives:

𝐺1(𝑢) = 𝒮[𝑓,(𝑡)] =

𝐺(𝑢)−𝑓(0)

𝑢

𝐺(𝑢)

𝑢

−

𝑓(0)

𝑢

𝐺2(𝑢) = 𝒮[𝑓,,(𝑡)] =

𝐺(𝑢) − 𝑓(0)

𝑢2 =

𝐺(𝑢)

𝑢2 −

𝑓(0)

𝑢2 −

𝑓,(𝑜)

𝑢

𝐺𝑛(𝑢) = 𝒮[𝑓𝑛(𝑡)] =

𝐺(𝑢)

𝑢𝑛 −

𝑓(0)

𝑢𝑛 − ⋯ −

𝑓𝑛−1(𝑜)

𝑢

Sumudu transform of integral of a function:

𝒮 [∫ 𝑓(𝜏)𝑑𝜏

0 ] = 𝑢𝐺(𝑢)

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SUMUDU TRANSFORM FUNDAMENTAL PROPERTIES

THE DISCRETE SUMUDU TRANSFORM

Over the set of functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}, (1)

the Sumudu transform is defined by

𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

(2)

Among others, the Sumudu transform was shown to have units preserving properties and

hence may be used to solve problems without resorting to the frequency domain. As will

be seen below, this is one of many strength points for this new transform, especially with

respect to applications in problems with

physical dimensions. In fact, the Sumudu transform which is itself linear, preserves linear

functions, and hence in particular does not change units (see for instance Watugala or

Belgacem et al).Theoretically, this point may perhaps best be illustrated as an implication

of this more global result.

THEOREM:1

The Sumudu transform amplifies the coefficients of the power series function,

𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 (1.1)

by sending it to the power series function,

𝐺(𝑢) = Σ 𝑛! 𝑎𝑛𝑢𝑛 ∞

𝑛

=0 (1.2)

PROOF :

Let f (t) be in A. If 𝑓(𝑡) = Σ 𝑎𝑛𝑡𝑛 ∞

𝑛

=0 in some interval I ⊂ R, then by Taylor’s

function expansion theorem,

𝑓(𝑡) = Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑡𝑛

(1.3)

Therefore, by (2), and that of the gamma function Γ , we have

𝒮[𝑓(𝑡)] = ∫ Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 (𝑢𝑡)𝑛𝑒−𝑡 ∞

0 𝑑𝑡

= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 ∫ 𝑡𝑛 ∞

0 𝑒−𝑡 𝑑𝑡

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= Σ 𝑓(𝑛)(𝑜)

𝑛!

∞𝑘

=0 𝑢𝑛 Γ(n+1)

= Σ 𝑓(𝑛)(0)𝑢𝑛 ∞

𝑛

=0 (1.4)

Consequently, it is perhaps worth noting that since

𝒮[(1 + 𝑡)𝑚] = 𝒮 Σ 𝐶𝑛

𝑚𝑡𝑛 𝑚𝑛

= 𝒮 Σ 𝑚!

𝑛!(𝑚−𝑛)!

𝑚𝑛

=0 𝑢𝑛

𝒮[(1 + 𝑡)𝑚] = Σ

𝑚!

(𝑚−𝑛)!

𝑚 𝑛=

𝑢𝑛

=Σ 𝑃𝑛

𝑚𝑢𝑛 𝑚𝑛

=0 (1.5)

the Sumudu transform sends combinations, 𝐶𝑛

𝑚 into permutations, 𝑃𝑛

𝑚, and hence may

seem to incur more order into discrete systems.

Also, a requirement that 𝒮[𝑓(𝑡)] converges, in an interval containing u=0, is provided by

the following conditions when satisfied, namely, that

(𝑖)𝑓(𝑛)(0) → 0 𝑎𝑠 𝑛 → ∞,

(𝑖𝑖) lim

𝑛→∞

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

𝑢| < 1 (1.6)

This means that the convergence radius r of 𝒮[𝑓(𝑡)] depends on the sequence

𝑓(𝑛)(0), since

𝑟 = lim

𝑛→∞

𝑓(𝑛+1)(0)

𝑓(𝑛)(0)

| (1.7)

Clearly, the Sumudu transform may be used as a signal processing or a detection

tool,especially in situations where the original signal has a decreasing power tail .

However, care must be taken, especially if the power series is not highly decaying. This

next example may instructively illustrate the stated concern. For instance, consider the

function

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𝑓(𝑡) = {

𝐼𝑛 (𝑡 + 1) 𝑖𝑓 𝑡 ∈ (−1,1]

𝑜 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(1.8)

Since 𝑓(𝑡) = Σ (−1)𝑛−1 ∞

𝑛

𝑡𝑛

⁄𝑛 the expect for u=0

𝒮[𝑓(𝑡)] = Σ (−1)𝑛−1(𝑛 − 1)! 𝑢𝑛 ∞

𝑛

=1 (1.9)

Diverges throught , because its convergens radius

𝑟 = lim

𝑛→∞

(−1)𝑛−1(𝑛−1)!

(−1)𝑛𝑛!

lim

𝑛→∞

𝑛

= 0 (1.10)

RELATION BETWEEN SUMUDU AND LAPLACE TRANSFORM

In our study, we use the following convolution notation: double convolution

between two continuous functions F (x, y) and G(x, y) given by

𝐹1(𝑥, 𝑦) ∗∗ 𝐹2(𝑥, 𝑦) = ∫ ∫ 𝐹1(𝑥 − 𝜃1, 𝑦 − 𝜃2)𝐹2(𝜃1, 𝜃2)𝑑𝜃1𝑑𝜃2

𝑥

𝑦

The single Sumudu transform is defined over the set of the functions

𝐴 = {(𝑓(𝑡)|∃𝑀, 𝜏1,𝜏2 > 0, |𝑓(𝑡)| < 𝑀𝑒|𝑡|/𝜏𝑗 , 𝑖𝑓 𝑡 ∈ (−1)𝑗 × [0, ∞))}

by 𝐺(𝑢) = 𝒮[𝑓(𝑡)] = ∫ 𝑓(𝑢𝑡)𝑒−𝑡 𝑑𝑡, 𝑢 ∈ (−𝜏1, 𝜏2) ∞

A sufficient condition for the existence of the Sumudu transform of a

function f is of exponential order, that is, there exist real constants

M > 0, 𝐾1, and 𝐾2 , such that |𝑓(𝑡, 𝑥)| ≤ 𝑀𝑒

𝑡

𝐾1

𝑥

𝐾2

EXISTENCE OF THE SUMUDU TRANSFORM

THEOREM:2

If f is of exponential order, then its Sumudu transform 𝒮[𝑓(𝑡, 𝑥)] = 𝐹(𝑣, 𝑢)exists and is

given by

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑒−

𝑡

𝑣

−

𝑥

𝑢𝑓(𝑡, 𝑥)

∞

𝑑𝑡𝑑𝑥

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where

𝑢

𝑖

𝜏

𝑣

𝜇

𝑖

𝜉

The defining integral for F exists at points

𝑢

𝑣

𝜇

𝑖

𝜏

𝑖

𝜉

in the right half plane

𝜇

𝐾1

𝐾2

PROOF:

Using

𝑢

𝑖

𝜏

𝑎𝑛𝑑

𝑣

𝜇

𝑖

𝜉

We can express 𝐹(𝑣, 𝑢) as

𝐹(𝑣, 𝑢) = ∫ ∫ 𝑓(𝑡, 𝑥)𝑐𝑜𝑠(

𝑡

𝜏

𝑥

𝜉

)𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥

−𝑖 ∫ ∫ 𝑓(𝑡, 𝑥)𝑠𝑖𝑛(

𝑡

𝜏

𝑥

𝜉

)𝑒

−

𝑡

−

𝑥

𝜇

∞

0 𝑑𝑡𝑑𝑥

Then for values of 1

𝜇

𝐾1

𝐾2

we have

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑐𝑜𝑠(

𝑡

𝜏

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

𝐾1

−

)𝑡+(

𝐾2

−

𝜇

)𝑥

∞

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

and

∫ ∫ |𝑓(𝑡, 𝑥)| |𝑠𝑖𝑛(

𝑡

𝜏

𝑥

𝜉

)| 𝑒

−

𝑡ɳ

−

𝑥𝜇

∞

𝑑𝑡𝑑𝑥 ≤ 𝑀 ∫ ∫ 𝑒

(

𝐾1

−

)𝑡+(

𝐾2

−

𝜇

)𝑥

∞

𝑑𝑡𝑑𝑥

≤ 𝑀 (

ɳ𝐾1

ɳ−𝐾1

) (

ɳ 𝐾2

𝜇− 𝐾2

)

which imply that the integrals defining the real and imaginary parts of F exist for value of

𝑅𝑒 (

𝑢

+ 1

𝜇

) > 1

𝐾1

𝐾2

, and this completes the proof.

Thus, we note that for a function f, the sufficient conditions for the existence of the

Sumudu transform are to be piecewise continuous and of exponential order.

We also note that the double Sumudu transform of function f(t, x) is defind by

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𝐹(𝑣, 𝑢) = 𝒮2[𝑓(𝑡, 𝑥); (𝑣, 𝑢)] =

𝑢𝑣

∫ ∫ 𝑒−(

𝑡

𝑣

𝑥

𝑢

)𝑓(𝑡, 𝑥)

∞

𝑑𝑡𝑑𝑥

where, 𝑠2 indicates double Sumudu transform and 𝑓(𝑡, 𝑥 ) is a function which can be

expressed as a convergent infinite series. Now, it is well known that the derivative of

convolution for two functions 𝑓 and 𝑔 is given by

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥) =

𝑑

𝑑𝑥

𝑓(𝑥) ∗ 𝑔(𝑥)𝑜𝑟 𝑓(𝑥) ∗

𝑑

𝑑𝑥

𝑔(𝑥)

and it can be easily proved that Sumudu transform is

𝒮 [

𝑑

𝑑𝑥

(𝑓 ∗ 𝑔)(𝑥); 𝑣] = 𝑢𝒮 [

𝑑

𝑑𝑥

𝑓(𝑥); 𝑢] 𝒮[𝑔(𝑥); 𝑢] or

= 𝑢𝒮[𝑓(𝑥); 𝑢]𝒮 [

𝑑

𝑑𝑥

𝑔(𝑥); 𝑢].

The double Sumudu and double Laplace transforms have strong relationships that may be

expressed either as

(𝐼) 𝑢𝑣𝐹(𝑢, 𝑣) = £2 (𝑓(𝑥, 𝑦); (

𝑢

𝑣

))

Or (𝐼𝐼) 𝑝𝑠𝐹(𝑝, 𝑠) = £2 (𝑓(𝑥, 𝑦); (

𝑝

𝑠

))

where £2 represents the operation of double Laplace transform. In particular, the double

Sumudu and double Laplace transforms interchange the image of sin(x + t) and cos(x + t).

It turns out that

𝑠2[sin(𝑥 + 𝑡)] = £2[cos(𝑥 + 𝑡) =

𝑢 + 𝑣

(1 + 𝑢)2(1 + 𝑣)2

And

𝑠2[cos(𝑥 + 𝑡)] = £2[sin(𝑥 + 𝑡) =

(1 + 𝑢)2(1 + 𝑣)2

REFERENCE:

1. G. K. Watugala, Sumudu Transform: a new integral transform to solve differential

equations and control engineering problems, Internat. J. Math. Ed. Sci.Tech. 24 (1993) 35-

43.

2.G. K. Watugala, The Sumudu transform for functions of two variables, Math.

JETIR2107602 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org e778

Engrg. Indust. 8 (2002) 293-302.

3. M.A.Asiru,Sumudu transform and the solution of integral equations of onvolution type,

International Journal of Mathematical Education in Science and Technology 32 (2001),

no. 6, 906–910.

4. Further properties of the Sumudu transform and its applications, International Journal of

Mathematical Education in Science and Technology 33 (2002), no.3, 441–449.

5. Classroom note: application of the Sumudu transform to discrete dynamic systems,

International Journal of Mathematical Education in Science and Technology 34 (2003),

no. 6,944–949.

Mathematics Department

Thursday 16 December 2021

AN INTRODUCTION OF SUMUDU TRANSFORM

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