Analytical investigations of the Sumudu transform and applications to integral production equations
Hindawi Publishing Corporation
Mathematical Problems in Engineering
FETHI BIN MUHAMMED BELGACEM
AHMED ABDULLATIF KARABALLI
SHYAM L. KALLA
The Sumudu transform, whose fundamental properties are presented in this paper, is little known and not widely used. However, being the theoretical dual to the Laplace transform, the Sumudu transform rivals it in problem solving. Having scale and unitpreserving properties, the Sumudu transform may be used to solve problems without resorting to a new frequency domain. Here, we use it to solve an integral productiondepreciation problem.

1. Introduction
In [6], a new integral transform, called the Sumudu transform defined for functions of
exponential order, is proclaimed. We consider functions in the set A, defined by
A = f (t)  ∃M, τ1, and/or τ2 > 0,
such that f (t) < Met/τj , if t ∈ (−1) j × [0, ∞) .
For a given function in the set A, the constant M must be finite, while τ1 and τ2 need
not simultaneously exist, and each may be infinite. Instead of being used as a power to
the exponential as in the case of the Laplace transform, the variable u in the Sumudu
transform is used to factor the variable t in the argument of the function f . Specifically,
for f (t) in A, the Sumudu transform is defined by
G(u) = S f (t) =
∞
0
∞
0
f (ut)e−tdt, 0 ≤ u < τ2,
f (ut)e−tdt,
−τ1 < u ≤ 0.
Albeit similar in expression, the two parts in the previous definition arise because in
the domain of f , the variable t may not change sign. For instance, if a function is defined
for nonnegative t, then G(u) is solely defined for nonnegative u, as exemplified in the
prototypical case of the function f (t) = √t. Here, τ1 is simply not needed, M can be
(1.1)
(1.2)
taken equal to 1, while τ2 is infinite. Entry 5 in Table A.1 shows that this function maps to
a multiple constant of itself by the Sumudu transform. Alternatively, see (1.12). On the
other hand, for u ∈ (−∞, 1/a), the Sumudu transform of eat is
1
G(u) = 1 − au .
While we are in agreement with most of the claims expounded by Watugala [6], we
think that this transform is not so new as proclaimed. The Sumudu transform is
connected to the smultiplied Laplace transform (see [5]). This however in no way diminishes
its importance or usefulness. In fact, we show that the Sumudu transform has deeper
connections with the Laplace transform than previously established. We also present many of
the new transform properties that make it uniquely qualified to address and solve some
applied problems, especially ones in which the units of the problem must be preserved.
The discrete analog of the Sumudu integral transform (1.2) is defined for power series
functions f (t) = k∞=0 aktk, having an interval of convergence containing t = 0, as follows:
∞
k=0
So, the linear function f (t) = a0 + a1t transforms to itself, G(u) = a0 + a1u = f (u).
However, the power series
transforms to the geometric series
f (t) =
∞
k=0
(−1)k (at)k = e−at
k!
G(u) =
∞
k=0
1
(−1)k(au)k = 1 + au ,
with u in (−1/a, 1/a).
Equations (1.4), (1.5), and (1.6) reveal that the Sumudu transform amplifies the
coefficients of the power series according to their order, without changing the initial units of the
series. Therefore, a signal with increasingly decaying higherorder coefficients an
transforms to another with much more prominent tail end. So, the power series of et which
converges throughout R transforms to the geometric series of 1/(1 − t) which converges
only in the interval (−1, 1). Moreover, the discrete version of the Sumudu transform gives
us the insight of how to obtain f (t) from G(u). We simply divide the coefficients of the
power series for G(u) by the respective n! value to obtain the power series for f (t).
While it is harder to compute at times, the integral transform in (1.2) is clearly much
more general than its discrete counterpart defined in (1.4). May they be of differential,
integral, or engineering control nature, the Sumudu transform can certainly treat all
problems that are usually treated by the wellknown and extensively used Laplace transform
defined for (s) > 0 by
F(s) = £ f (t) =
e−st f (t)dt.
∞
0
Indeed, as the next theorem shows, the Sumudu transform is closely connected with the
Laplace transform.
Theorem 1.1. Let f (t) ∈ A with Laplace transform F(s). Then the Sumudu transform G of
f (t) is given by
Proof. Let f (t) ∈ A, then for −τ1 < u < τ2,
If we set w = ut (t = w/u), then the righthand side can be written as
The integral on the righthand side is clearly F(1/u), thus yielding (1.8).
We observe that G(1) = F(1) so that both the Sumudu and Laplace transforms must
coincide at u = s = 1. Furthermore, since for x > 0, the Gamma function
is the Laplace transform of tx−1(£(tx−1)) when s = 1, then Γ(x) must also be the Sumudu
transform (S(tx−1)) when u = 1. Indeed, multiplying the integral in (1.11) by ux−1 yields
the following result.
Corollary 1.2. For x > 0, the Sumudu transform of tx−1 is
G(u) = S tx−1
= Γ(x)ux−1.
In fact, the connection of the Sumudu transform with the Laplace transform goes
much deeper. Therefore, the roles of F and G in (1.8) can be interchanged.
Corollary 1.3. Let f (t) ∈ A, having F and G for Laplace and Sumudu transforms,
respectively. Then
F(s) =
Proof. Equation (1.13) can be obtained from (1.8) by taking u = 1/s.
The pair of equations (1.8) and (1.13) forms the duality relation governing these two
transforms and may serve as a means to get one from the other when needed. Following
the style of Kreyszig [4], Table A.1 shows both Laplace and Sumudu transforms of some
elementary and special functions. Table A.2 summarizes the properties of the Sumudu
transform as expanded upon below.
2. Sumudu transforms of derivatives and integrals
Let f (t) ∈ A and let F(s) = £( f (t)), the Laplace transform of f (t) with respect to s, and
let G(u) = S( f (t)); then
S[sin t] = 1 +uu2 ,
Being a restatement of the duality relation (1.8), (2.1) will serve as our working definition
throughout the paper. Since the Laplace transform of sin t is 1/(1 + s2), then in view of
(2.1), its Sumudu transform is
which is the Laplace transform of cos t (with u = s). This exemplifies the duality between
these two transforms, and hence emphasizes the importance of the Sumudu transform.
Obviously, the Sumudu transform is linear since the Laplace transform is. The next few
theorems are designed to illustrate how the Sumudu transform behaves with derivatives
and antiderivatives.
Theorem 2.1. Let F1(u) and G1(u) be the Laplace and the Sumudu transforms of the
derivative of f (t) (∈ A). Then
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
Clearly, from (2.2), being the derivative of sin t, relation (2.3) prescribes the expected
Sumudu transform for cos t:
Proof. Since the Laplace transform of the derivative of f (t) is
then
or
Theorem 2.2. Let n ≥ 1, and let Gn(u) and Fn(u) be the Sumudu and Laplace transforms
of the nth derivative f (n)(t), of the function f (t), respectively. Then
Proof. By definition, the Laplace transform for f (n)(t) is given by
Therefore,
Now, since Gk(u) = Fk(1/u)/u, for 0 ≤ k ≤ m, we have
Gn(u) =
In particular, this means that the Sumudu transform of the second derivative of the
function f is given by
G2(u) = S f (t) = Gu(2u) −
For instance, applying (2.12) to the function sin t leads us to the equation
whose solution is obviously the Sumudu transform of sin t, given by (2.2).
Theorem 2.3. Let G1(u) and F1(s) denote the Sumudu and the Laplace transforms of the
definite integral of f , W (t) = 0t f (τ)dτ, respectively. Then
Proof. By definition, the Laplace transform of W (t) = 0t f (τ)dτ is given by
−G(u) =
G(u) 1
u2 − u
G1(u) = S W (t) = uG(u).
F1(s) = £ W (t) =
F(s)
s
Now, recall that the antiderivative of the Dirac delta function, δ(t − a) (see, e.g., [4]),
is the Heaviside function H(t − a) defined by
H(t − a) =
0, if t < a,
1, if t > a.
S H(t − a) = e−a/u.
Knowing that the Sumudu transform of δ(t − a) is e−a/u/u (see entries 34 and 35 in
Table A.1), using (2.14), yields the transform of H(t − a):
Another facet of the duality relation between this transform and the Laplace transform is
revealed through the interchange of the images of H(t) and δ(t).
Note that the Sumudu transforms of H(t) and δ(t) are 1 and 1/u, respectively. This is
consistent with the units and the buildup of these functions (see, e.g., [4, Section 6.4]).
Now, the Dirac δ(t) is really a generalized function. Thus, recalling (2.14), we may be able
to make sense, in a generalized fashion, of the notion of a derivative for δ(t). Indeed, if a
generalized function g(t) were to exist such that
then, we must have
t
0
t
0
S
g(τ)dτ
1
= u ,
g(τ)dτ = δ(t).
S f (at) = G(au).
This result will be useful in Section 4. Next we establish the scalepreserving property of
this transform.
Theorem 2.4. Let f (t) ∈ A with Laplace and Sumudu transforms F(s) and G(u),
respectively. Then
Proof. The Sumudu transform of f (at) may be obtained directly from the definition
(1.2):
or via the working defining equation (2.1). Indeed, since (see, e.g., [4])
∞
0
S f (at) =
f (aut)e−tdt = G(au)
1
£ f (at) = a F
s
a
,
then
S f (at) =
(1/a)F(1/au)
u
=
F(1/au)
au
Theorem 2.5. Let f ∈ A with Sumudu transform G(u). Then,
(ut) f (ut)e−tdt = u1 S t dfd(tt)
Multiplying both sides by u, we get the desired result in (2.25).
We observe that with the Sumudu transform, differentiation and integration in the
tdomain are akin to division and multiplication in the udomain, respectively.
Furthermore, many of the scaling properties of f (t) are carried over to its Sumudu transform
G(u). Hence both f (t) and G(u) keep the same units, and u and G(u) can be treated as
replicas of t and f (t), respectively. This is a major advantage in transform theory,
especially when dealing with applications, where being aware of the units of the quantities
described as well as the dimensionless factor groups may be extremely relevant in
problem solving.
In view of these advantages and the duality relation between the Sumudu and Laplace
transforms, there may be applied situations where using the Sumudu transform may be
favored over using the Laplace transform. This will be illustrated at length in an upcoming
paper dealing with Brownian motion and weighted convection diffusion equations (see
[1, 2]). In the meantime, we take advantage of the duality to investigate more properties
of the Sumudu transform.
3. More properties of the Sumudu transform
The next few theorems establish some translation, shift, and limit properties of the
Sumudu transform.
Theorem 3.1. Let f (t) ∈ A with Sumudu transform G(u). Then,
Proof. From (1.2), we see that 1
Therefore, by a change of variable (w = (1 − au)t), we get that
1 −uwau e−wdw = 1 −1au G
.
Theorem 3.2. Let f (t) ∈ A with Laplace and Sumudu transforms F(s) and G(u),
respectively. Then the function
h(t) =
0,
f (t − a), if t > a,
if t < a,
S h(t) = e−a/uG(u).
£ h(t) = e−asF(s).
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
Theorem 3.4. Let f (t) ∈ A and suppose that either limt→0 f (t) or limt→∞ f (t) exists. Then
has a Sumudu transform given by
Proof. Note that from (2.17), h(t) = H(t − a) f (t − a), and hence the Laplace transform
of h(t) is given by
Therefore, by duality, the Sumudu transform of h(t) is given by
S h(t) = e−a/u F(1/u) = e−a/uG(u).
u
The next theorem shows that the average of f over [0, t] transforms to the average of
G over [0, u].
Theorem 3.3. Let f (t) ∈ A with Sumudu transform G(u). Then,
S 1 t
t 0
f (τ)dτ
G(v)dv.
1 u
= u 0
Proof. From definition (1.1), we have
u1 0u G(v)dv = u1 0u 0∞ f (vt)e−tdt dv = u1 0∞ e−t
= 0
= 0 ut
∞ e−t u
u
∞ 1
0
f (w) dtw dt =
f (w)dw
e−tdt =
∞ 1
e−t
0 ut 0
S 1 t
t 0
ut
0
lim G(u) = lti→m0 f (t),
u→0
lim G(u) = lim f (t).
u→∞ t→∞
∞
0
ut
f (vt)dv dt
f (w)dw dt
f (τ)dτ .
Proof. The first limit is obtained as follows: In the same manner,
∞
limG(u) = lui→m0 0 f (ut)e−tdt =
u→0
∞
0
lim f (ut) e−tdt
u→0
∞
lim f (w) e−tdt = wli→m0 f (w)
w→0
e−tdt = wli→m0 f (w).
∞
0
∞
0
∞
lim G(u) =
u→∞
lim f (ut) e−tdt
u→∞
lim f (w) e−tdt = wli→m∞ f (w).
w→∞
A similar argument yields the negative counterpart to (3.11):
lim G(u) = lim f (t),
u→−∞ t→−∞
when the righthand side of (3.14) exists.
The results indicated in (3.10) and (3.11) are known to be the initial and final value
theorems, respectively. The reader can observe that most of the previous proofs may also
be obtained by the duality relation (2.1). For instance, (3.10) and (3.11) can alternatively
be gotten as follows:
lim G(u) = lui→m0
u→0
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
Similarly,
Note that
Proof. The Laplace transform of the periodic function f (t) is given by
S f (t) = 0T/u e−t f (ut)dt
.
1 − e−T/u
£ f (t) =
0T e−st f (t)dt
1 − e−sT
.
T
e−st f (t)dt = u
e−t f (ut)dt
T/u
and that, in particular,
1 − e−T/u = 0
T/u
e−tdt.
Therefore, by definition (2.1), we have
4. Applications to an integral production problem
In this section, we use the Sumudu transform to treat a productiondepreciation
problem, first considered by Kalla et al. [3]. This problem examines the manner the production
of an item varies in time, when for known losses due to depreciation, the total amount of
the product is to remain constant. The following model can easily be adopted to weighted
populationgrowth models (see [1]), birthdeath processes, and hormonal and drug
release control problems.
If at t = 0, the unused amount of a product is M, despite the exposure of the amount
M of the product to a depreciation function f (t), for t ≥ 0, we like the production of this
item g(t) to counterbalance the depreciation so as to keep the amount M at all times.
Note that there is no loss of generality in assuming that the depreciation function over
time satisfies the condition
f (t)dt = 1.
M f (t)dt = M.
∞
0
∞
0
t
Hence, the amount lost due depreciation in the absence of production (g(t) = 0, t > 0) is
given by
The amount of the item produced in the absence of depreciation ( f (t) ≡ 0) in an interval
of time [x, x + ∆ x] is given by g(x)∆ x. When depreciation takes place, the amount of the
item lost at a later time t is given by g(x) f (t − x)∆ x, and the total loss due to depreciation
from the start until time t is given by the convolution integral
t
0
Therefore, if both production and depreciation are simultaneously in effect by the time
t, the net difference of production and loss must equal M f (t). That is,
t
Kalla et al. [3] solved (4.4) for the depreciation function
f (t) = Γ(a(k)k) tk−1e−at,
with constants a and k (see entry 10 in Table A.1), and obtained the production function
g(t) = M
∞ n!ank
n=1 Γ(nk)
tnk−n−1e−atLnnk−n−1(−at),
where Lnp(x) are the generalized Laguerre polynomials.
To solve the integral equation (4.4) our way, we first introduce the Sumudu transform
convolution theorem.
Theorem 4.1. Let f (t) and g(t) be in A, having Laplace transforms F(s) and G(s),
respectively, and Sumudu transforms M(u) and N (u), respectively. Then the Sumudu transform
of the convolution of f and g,
the Sumudu transform of ( f ∗ g) is obtained as follows:
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
is given by
and since
Proof. First, recall that the Laplace transform of ( f ∗ g) is given by
Now, since, by the duality relation,
∞
0
S ( f ∗ g)(t) =
F(1/u) × G(1/u)
u
F(1/u) G(1/u)
= u u u
= uM(u)N (u).
Now, to solve our production problem, letting G(u) and F(u) be Sumudu transforms
of the sought production function g(t) and of the depreciation function f (t),
respectively, (4.4) becomes
Therefore, we have
Let f (t) = e−t, then
Consequently, from (4.16),
uG(u) − uG(u)F(u) = MF(u).
MF(u)
G(u) = u 1 − F(u) .
Therefore, by linearity of the Sumudu transform (2.19) and (2.20), we deduce that
Now, in light of (4.5) and (4.6), our depreciation function coincides with that of Kalla
et al. [3], when a = k = 1. Hence, our solution g(t) must agree with theirs for the same
values. For f (t) = e−t, we take the production function
where Ln−1 are the Laguerre generalized polynomials. On the other hand, g(t) is to satisfy
(4.18). Therefore, we must have
Whence, if there is a notion of a generalized derivative for δ(t), then g(t), as defined in
(4.19), is a most likely suspect.
Appendix
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
f (t)
1
f (t)
1
2ω (sin ωt + ωt cos ωt)
1
b2 − a2 (cos at − cos bt), a2 = b2
1
4k3 (sin kt cosh kt − cos kt sinh kt)
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
21k2 sin kt sinh kt
1
2k3 (sinh kt − sin kt)
1
2k2 (cosh kt − cos kt)
1
2√πt3 ebt − eat
e−(a+b)t/2I0 a − b t
2
J0(at)
1
√πt eat(1 + 2at)
√π t k−1/2
Γ(k) 2a
H(t − a)
δ(t − a)
J0 2√kt
√1πt cos 2√kt
√1πt sinh 2√kt
k
2√πt3 e−k2/4t, (k > 0)
ln t + γ (γ 0.5772 . . .)
[1]
EditorinChief
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USA
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Fethi Bin Muhammed Belgacem : Faculty of Information Technology, Arab Open University, Block 3, Street 7 , No. 37, P.O. Box 3322 , Safat 13033 , Kuwait E mail address: fbmbelgacem@yahoo . com Ahmed Abdullatif Karaballi: Department of Mathematics and Computer Science , Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060 , Kuwait E mail address: .kuniv .edu.kw Shyam L. Kalla: Department of Mathematics and Computer Science, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060 , Kuwait E mail address: Volume 2014 Volume 2014 Volume 2014