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A Study of Triple Sumudu Transform for Solving
Partial Differential Equations with Some Applications
Mohammed S. Mechee
Abbas J. Naeemah
University of Kufa, Najaf, Iraq
Abstract. In this paper, we have studied the extension of the single Sumudu
transform to deal with the functions in three variables. Furthermore, we have studied the
main properties of triple Sumudu transform. In addition, we have applied triple Laplace
transform with its main characteristics for solving several examples. Finally, we have
examined the solutions for general second-order PDEs which containing three variables
by using triple Laplace and Sumudu transforms.
Key words: Sumudu Transform; Laplace Transform; Triple; Partial Differential
Equations.
Introduction
Partial differential equation (PDEs) plays a very important role in mathematics and
the other fields of sciences because these linear or nonlinear PDEs describe the physical
phenomena. Thus, it is important to know how to solve these PDEs. A number of
numerical or analytical methods of solutions can be used to find the solutions of DEs. The
numerical methods have provided the approximated solution of differential equations
(DE) rather than the analytical solutions of the problems of the study. In most times it may
be difficult to solve these DEs analytically and thus are commonly solved by integral
transforms such as Laplace and Fourier transforms and the advantage of these two
methods lies in their ability to transform DEs into algebraic equations, which allow a
simple way to find the solutions. In Eltayeb and Kiliçman (2013), Eltayeb et al. (2012: 47);
Kiliçman and Eltayeb (2008: 1124), Kiliçman and Gadain (2010) extended the concept of
Laplace transform to double Laplace transform and this new operator has been widely
used to solve some kinds of DEs. Then, the concept of triple Laplace transform was used
to solve third-order PDEs. In addition, the properties and the applications to DEs have
been determined and studied (Atangana, 2013; Khan et al., 2019; Shiromani, 2013: 848).
As we see, the integral transform method is an effective way to solve some certain DEs.
Thus, in the literature there are a lot of works on the theory and applications of Laplace,
Fourier, Mellin and other integral transforms (Debnath and Bhatta, 2006). A little on the
power series transformation such as Sumudu transform, maybe because it is little known
and not widely used yet. Sumudu transform was proposed by Watugala (1993: 35) for
solving DEs and control engineering problems. Among the other integral transforms,
Sumudu transform has units preserving properties and thus may be used to solve
problems without resorting them to the frequency domain and this is one of many strength
points of this new transform. However, Belgacem et al. (2003: 103) extended the theory
and the applications of Sumudu transform and applied it for fractional integrals,
derivatives, and used it to solve initial value fractional differential equations (FrDEs) and
(Asiru, 2002: 441) further developed this new transform and most of its fundamental
properties and applied it for special functions and used it to solve fractional integrals,
derivatives, and initial value FrDEs. Since then, many researchers have studied Sumudu
transform and its properties. Series of papers have been published started with Belgacem
and Karaballi (2006), where he extended the theory and the applications of the Sumudu
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transform and use it to solve the FrDEs by direct integration methods, study and prove
most of Sumudu transform properties study the Laplace-Sumudu transforms duality and
the complex inversion formula and avails the readers with the most comprehensive list of
function transforms in the literature, up to date Kiliçman and Gadain (2010: 10) studied
the extension of double Sumudu transform to triple Sumudu transform briefly and study
some of its properties and its relation with triple Laplace transform. Haydar (2009: 33)
studied the extension of single Sumudu transform to n-dimensions Sumudu transform
and studied its main properties and gives a table of n-dimensions Sumudu transform for
the most familiar functions. In this paper, we have studied the extension of the concept
of Sumudu transform into triple Sumudu transform along with its main properties, studied
triple Laplace transform with its main characteristics and explained them by several
examples.
Preliminary
Here, we give the definition for triple Sumudu transform with the properties of triple
Sumudu and Laplace transforms.
The Triple Sumudu Transform
In this subsection, we give the definition for triple Sumudu transform with the
properties of triple Sumudu and Laplace transforms.
Definition. The triple Sumudu transform can be defined by:
or by,
Properties of the Triple Sumudu Transform
1. The triple Sumudu transform of the third partial derivative with respect to x
has the following form:
The integral inside the brackets can be computed individually as follows:
2. By taking Sumudu transform with respect to y, we get the double Sumudu
transform as follows:
3. By taking Sumudu transform with respect to t ,we get the double Sumudu
transform as follows:
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4. The triple Laplace transform was defined by the following form:
5. The triple Laplace transform for the third-order partial derivative with respect
to x can be given by:
6. By the same way, the triple Laplace transform for the third-order partial
derivative with respect to y can be given by
7. By the same way, the triple Laplace transform for the third-order partial
derivative with respect to t can be given by
Results
In this section, we have applied the triple Sumudu and Laplace transforms for
solving general linear third- and fourth-orders PDEs.
General Linear Third-Order PDEs
In this section, the following general third-order PDE has been considered.
and .
Triple Laplace Transform
In this subsection, Laplace transform for solving Equation (2) has been used with
assumptions. a7 = a8 = a9 = a10 = a11 = a12 = a13 = a14 = 0 and f(x;y;t) = 0 as follows
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Now, by taking triple Laplace transform of expanded ICs as follows
and,
Substitute equations (2.3) -(2.11) in Equation (3) to obtain the following
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In particular, if a1 = a2 = a3 = a4 = a5 = a6 = a15 = a161 = a17 -a18 = 1; we obtain
Where
Then,
Using inverse of Laplace transform for Equation (13), we obtain the following
solution
Example 1. Consider the following third-order PDE
With IC:
BC:
Using Laplace transform for Equation (14)
Where
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Then,
Taking inverse of Laplace transform for Equation (15) to obtain the following
solution:
Triple Sumudu Transform
In this subsection, the triple Sumudu transform for solving PDEs with three variables
Equation has been used (2) with assumption . a7 = a8 = a9 = a10 = a11 = a12 = a13 =
a14 = 0 and f (x;y;t) = 0 as follows:
Taking triple Sumudu transform of ICs
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and,
Substitute equations (2.22) -(2.30) into Equation (16) to obtain the following
In particular, if
Where
Using inverse of Sumudu transform for Equation (26) to obtain the following solution:
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Example 2. Consider the following third-order PDE
With IC
BC:
Taking Sumudu transform for Equation (27), to obtain the following solution:
Where,
Hence,
Using Sumudu inverse transform for Equation (28) to obtain the following solution:
General Linear Fourth-Order PDEs
In this section, the following general fourth-order PDE has been considered
Where, are constants.
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with IC:
BC:
Laplace Transform
In this subsection, we use triple Laplace transform for solving Equation (29) with the
following assumptions. a7 = a8 = a9 = a10 = a11 = a12 = a13 = a14 = 0 = a18 = a19 =
a20 = a21 = a22 = a23 = 0 and f(x;y;t) = 0 as follows
Taking triple Laplace transform of ICs
So,
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Where,
Substitute equations (3.3) -(3.14) into Equation (30) to obtain the following.
And,
In particular, if a1 = a2 = a3 = a4 = a5 = a6 = a15 = a16 = a17 = a24 = a25 = a26-
14a18 = 1
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Using Laplace inverse transform for Equation (44) to obtain the following solution:
Example 3. Consider the following Fourth-order PDE
with IC:
BC:
Using Laplace transform for Equation (45), to obtain the following solution:
Taking Laplace inverse transform for Equation (47) to obtain the following solution:
Triple Sumudu Transform
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In this subsection, we use triple Sumudu transform for solving fourth-order PDEs in
general in Equation (29) with the following assumptions . a7 = a8 = a9 = a10 = a11 = a12
= a13 =a14 = 0 = a18 = a19 = a20 = a21 = a22 = a23 = 0 and f (x;y;t) = 0 as follows:
Applying triple Sumudu transform of ICs
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and
Substitute equations (3.21) -(2.32) into Equation (48) to obtain the following solution:
In particular, if a1 = a2 = a3 = a4 = a5 = a6 = a15 = a161 = a17a24 = a25 = a26 -
14a18 = 1, then,
Where
Then,
Using Sumudu inverse transform for Equation (61) to obtain the following solution
Example 4. Consider the following fourth-order PDE
With IC
BC:
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Where,
Using Sumudu inverse transform for Equation (63) to obtain the following solution:
Conclusion
In this paper, the triple Sumudu transform has been studied. The properties of the triple
Sumudu transform have been derived. The triple Sumudu transform used for solving
some PDEs problems. The approximated solutions of these problems using this
transform agrees very well with the analytical solutions. As such, this method is more cost
effective in terms of computation steps than other existing transform methods. Hence, we
can conclude that the new method is computationally very efficient in solving PDEs.
Acknowledgement
The authors would like to thank University of Kufa for supporting this research
project in part funding.
References
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