Solution of the Abel integral equation is obtained using the Sumudu transform and further, distributional Sumudu transform, and, distributional Abel equation are established

Deshna Loonker

P. K. Banerji

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International Journal of Mathematics and Mathematical Sciences

On the Solution of Distributional Abel Integral Equation by Distributional Sumudu Transform

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Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 480528, 8 pages
doi:10.1155/2011/480528
Research Article
On the Solution of Distributional Abel Integral
Equation by Distributional Sumudu Transform
Deshna Loonker and P. K. Banerji
Department of Mathematics, Faculty of Mathematics, Jai Narain Vyas University, Jodhpur 342 005, India
Correspondence should be addressed to Deshna Loonker, deshnap@yahoo.com
Received 20 October 2010; Accepted 8 January 2011
Academic Editor: Christian Corda
Copyright q 2011 D. Loonker and P. K. Banerji. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Solution of the Abel integral equation is obtained using the Sumudu transform and further,
distributional Sumudu transform, and, distributional Abel equation are established.
1. Introduction
This section deals with the definition, terminologies, and properties of the Sumudu transform
and the Abel integral equation. In Section 2, solution of Abel integral equation is obtained by
the application of the Sumudu transform, and in Section 3, the Sumudu transform is proved
for distribution spaces, and the solution of Abel integral equation in the sense of distribution
is obtained.
The Sumudu transform is introduced by Watugala 1, 2 to solve certain engineering
problems. Complex inversion formula for the Sumudu transform is given by Weerakoon 3.
For more of its applications, see 4–6.
Let M be a constant and τ1 and τ2 need not exist simultaneously each may be
infinite. Then, the set A is defined by
A 
{
ft | Me|t|/τj ; if t ∈ −1j × 0,∞
}
, 1.1
which initiates the definition of the Sumudu transform, see 4, in the form
Fu  S
[
ft
]

⎧
⎪⎪⎨
⎪⎪⎩
∫∞
0
fute−tdt; 0 ≤ u < τ2,
∫∞
0
fute−tdt; τ1 < u ≤ 0.
1.2
2 International Journal of Mathematics and Mathematical Sciences
In other words, the Sumudu transform can also be written as 4, 6
Fu  S
[
ft;u
]

1
u
∫∞
0
e−t/uftdt, u ∈ −τ1, τ2 1.3
inversion formula of which is given by
ft 
1
2πi
∫αi∞
α−i∞
Fuet/udu. 1.4
The discrete analog of the Sumudu transform 1.2 for the power series function ft ∑∞
k0 akt
k, having an interval of convergence containing t  0, is given by
Fu 
∞∑
k0
k!akuk, u ∈ τ1, τ2. 1.5
The Laplace-Sumudu duality is expressed as
F
(
1
s
)
 sLs, L
(
1
u
)
 uFu, 1.6
where F is the Sumudu transform and L is the Laplace transform.
The Sumudu transform of nth order derivative of ft is defined by
S
[
f nt
]
 Fnu 
Fu
un
−
n−1∑
k0
f k0
un−k
. 1.7
Convolution of the Sumudu transform is
S
(
f ∗ g)t  uS[ft]S[gt]  uFuGu. 1.8
Fnu denotes the Sumudu transform of nth antiderivative of ft, which is obtained by
integrating the function ft n times successively, that is
Fnu  unFu. 1.9
If Fu is the Sumudu transform of a function ft in A, f nt is the nth derivative of
ftwith respect to t, and Fnu is the nth derivative of Fuwith respect to u, then the Sumudu
transform of the function is
S
[
tnf nt
]
 un
dnFu
dun
 unFnu. 1.10
Some properties of the Sumudu transform, see 6, relevant to present paper may be
considered as below.
International Journal of Mathematics and Mathematical Sciences 3
Lemma 1.1. Let ft and gt be continuous functions defined for t ≥ 0, possessing Sumudu
transforms Fu and Gu, respectively. If Fu  Gu almost everywhere, then ft  gt, where
u is a complex number.
Theorem 1.2 existence of Sumudu transform. If f is of exponential order, then, indeed, its
Sumudu transform Fu exists, which is given by
Fu 
1
u
∫∞
0
fte−t/udt, 1.11
where 1/u  1/η  i/τ . The defining integral for F exists at point 1/u  1/η i/τ in the right hand
plane η > K and ζ > L.
Proposition 1.3 Sumudu transform of higher derivatives. Let f be n times diﬀerentiable on
0,∞ and ft  0 for t < 0. Further, suppose that f n ∈ Lloc. Then f k ∈ Lloc for 0 ≤ k ≤ n − 1,
domSf ⊂ domSf n, and for any polynomial P of degree n
PuS
(
y
)
u  S
(
f
)
u Mpuφ
(
y, n
)
1.12
for u ∈ domSf. In particular,
(
Sf n
)
u 
1
un
(
Sf
)
u
−
(
1
un
,
1
un−1
, . . . ,
1
u
)
φ
(
f ;n
)
, 1.13
where by φf ;n one means a column vector and dom will mean domain.
The Abel integral equation is given by 7, page 43
fs 
∫ s
a
gt
s − tα dt, s > a, 0 < α < 1, 1.14
solution of which is given by
gt 
sinαπ
π
d
dt
[∫ t
a
fs
t − s1−α
ds
]
. 1.15
The solution can be obtained by two methods which are shown in 7, pages 44-45.
2. Solution of Abel Integral Equation Using Sumudu Transform
In this section, we prove the Abel integral equation using the Sumudu transform. We write
the Abel integral equation in the form
ft 
∫ t
0
gs
t − sα ds, 0 < α < 1, 2.1
4 International Journal of Mathematics and Mathematical Sciences
which can also be written as
ft  gt ∗ t−α , 2.2
where t−α  t
−αHt,Ht is Heavside’s unit step function. Applying the Sumudu transform
on both sides of 2.2 and using the convolution of the Sumudu transform 1.8 in 2.2, we
have
S
[
ft
]
 uS
[
gt
] ∗ S[t−α
]
. 2.3
When ft  tn−1/n − 1!, the Sumudu transform is Fu  un−1. Similarly, we prove that if
ft  t−α, then the Sumudu transform is Fu  Γ1 − αu−α, where Ht  1, t ≥ 0. Putting
value of St−α in 2.3, we have
Fu  uS
[
gt
] · Γ1 − αu−α,
S
[
gt
]

Fu
Γ1 − α ·
uα
u

FuΓα
ΓαΓ1 − α ·
uα
u
; ΓαΓ1 − α  π
sinπα

Γα sinπα
π
· Fuu
α
u

Γα sinπα
π
· Fuuα−1 .
2.4
That is,
S
[
gt
]

sinπα
π
[
ΓαFuuα−1
]

sinπα
π
S
[
tα−1 ∗ Fu
]
, S
[
tn−1
]
 Γnun−1
S
[
gt
]

sinπα
π
S
[∫ t
0
t − sα−1fsds
]
.
2.5
That is,
S
[
gt
]

sinπα
π
SKt, 2.6
where Kt 
∫ t
0 t − sα−1fsds,K0  0.
International Journal of Mathematics and Mathematical Sciences 5
By virtue of 1.7, SK′t  SKt −K0/u  SKt/u. Invoking it in 2.6, we
have
S
[
gt
]

sinπα
π
uS
[
K′t
]
, 2.7
that is,
gt 
sinπα
π
u
dKt
dt
, 2.8
that is,
gt 
sinπα
π
u
d
dt
[∫ t
0
t − sα−1fsds
]
. 2.9
This is the required solution of 2.1, the Abel integral equation, by virtue of the Sumudu
transform.
3. Sumudu Transform and Abel Integral Equation on
Distribution Spaces
This section deals with the Sumudu transform on certain distribution spaces, and,
subsequently a relation is established to solve the Abel integral equation by the distributional
Sumudu transform.
If ft is a locally integrable function, then the distribution f through the convergent
integral ft in D′ is defined by
〈
f, φ
〉

〈
ft, φt
〉
 
∫∞
−∞
ftφtdt. 3.1
By virtue of Proposition 1.3, the Sumudu transform of the function ft generates a
distribution, or in other words, ft is in D′ and φt belongs to D, where D and D′ denote,
respectively, testing function space and its dual.
The linearity property, defined in 4, page 117, is
S
[
aft  bgt
]
 aS
[
ft
]
 bS
[
gt
]
, 3.2
where a and b are any numbers.
By virtue of 3.1 and 3.2, we state, if the locally integrable functions ft and gt
are absolutely integrable over 0 < t < ∞ and if their Sumudu transforms Fu and Gu are
equal everywhere, then ft  gt almost everywhere.
In what follows is the proof of the Parseval equation for the distributional Sumudu
transformation, which will be employed in the analysis of the problem of this paper.
6 International Journal of Mathematics and Mathematical Sciences
Theorem 3.1. If the locally integrable functions ft and gt are absolutely integrable over 0 < t <
∞, then
∫∞
0
FuGudu 
∫∞
0
ftg−tdt. 3.3
Proof. Since the transforms Fu and Gu are bounded and continuous for all u, as shown in
Section 1, therefore both the sides of 3.3 converge. Moreover,
∫∞
0
FuGudu 
1
u
∫∞
0
fte−t/udt
∫∞
0
Gudu

1
u
∫∞
0
ftdt
∫∞
0
Gue−t/udu.
3.4
Since the above integral is absolutely integrable, therefore
∫∞
0
FuGudu 
∫∞
0
ftg−tdt. 3.5
Further, we consider gt  f∗−t such that
∫∞
0
FuF∗udu 
∫∞
0
ftf∗tdt, 3.6
that is,
∫∞
0
|Fu|2du 
∫∞
0
∣∣ft∣∣2dt. 3.7
Thus, the Parseval relation of the Sumudu transform can be written as
‖F‖  ∥∥f∥∥. 3.8
This proves the theorem.
It may not be out of place to mention that 1.14 can be attained by virtue of
the convolution of distribution for explicit interpretation, refer to 7, page 180. Consider
convolution as a bilinear operation ∗ : D′41a,∞×D′41b,∞ → D′41ab,∞. If u ∈ D′41a,∞
and v ∈ D′41b,∞ are locally integrable functions, then we have
u ∗ vt 
∫ t−b
a
uτvt − τdτ, t > a  b. 3.9
International Journal of Mathematics and Mathematical Sciences 7
When b  0 and v ∈ D′410,∞, we have u ∗ v ∈ D′41a,∞. Thus, the convolution with
v defines an operator of the space D′41a,∞, which is given by
u ∗ vt 
∫ t
a
uτvt − τdτ, t > a, 3.10
where u and v are locally integrable functions.
The convolution form of 1.14 is
f  g ∗ t−α , 3.11
where t−α  t
−αHt and t−α is locally integrable, since 0 < α < 1.
Equation 3.11 asserts that the Abel integral equation can be interpreted in the sense
of distributions and the functions f and g can be considered to be elements of D′41a,∞.
Similarly, 1.15 can also be interpreted in the sense of distribution, given by
gt 
sinαπ
π
d
dt
(
f ∗ tα−1
)
. 3.12
It may be remarked that the Sumudu transform has an aﬃnity for the mixed spaces,
by virtue of which it is identified, owing to the fact that D′41a,∞, and similarly others
mentioned above, is one of the mixed distribution space that is identified with the space
of distribution D′R, support of which is contained in a,∞.
Whereas 3.11 and 3.12 express the solution of Abel integral equation on certain
distribution spaces, the similar method is invoked as in Section 2 to obtain the solution of
the Abel integral equation by using the distributional Sumudu transform. The analysis is,
therefore, explicitly explained and justified.
Acknowledgments
This paper is partially supported by the DST SERC, Government of India, Fast Track
Scheme Proposal for Young Scientist, no. SR/FTP/MS-22/2007, sanctioned to the first author
D. Loonker and the Emeritus Fellowship, UGC India, no. F.6-6/2003 SA-II, sanctioned
to the second author P. K. Banerji. The authors are thankful to the concerned referee for
fruitful comments that improved their understanding.
References
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3 S. Weerakoon, “Application of Sumudu transform to partial diﬀerential equations,” International
Journal of Mathematical Education in Science and Technology, vol. 25, no. 2, pp. 277–283, 1994.
4 F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform
and applications to integral production equations,” Mathematical Problems in Engineering, no. 3-4, pp.
103–118, 2003.
8 International Journal of Mathematics and Mathematical Sciences
5 F. B. M. Belgacem and A. A. Karaballi, “Sumudu transform fundamental properties investigations
and applications,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 91083, 23
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of Mathematical Analysis, vol. 4, no. 1–4, pp. 123–132, 2010.
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On the Solution of Distributional Abel Integral Equation by Distributional Sumudu Transform

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