Journal ArticleDOI

Application of Sumudu transform to partial differential equations

01 Apr 1994-International Journal of Mathematical Education in Science and Technology (Taylor & Francis Group)-Vol. 25, Iss: 2, pp 277-283
TL;DR: In this paper, the Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations (PDEs) is demonstrated with respect to three different PDEs.
Abstract: The Sumudu transform of partial derivatives is derived, and its applicability demonstrated using three different partial differential equations.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the Sumudu transform is used to solve problems without resorting to a new frequency domain, which is the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving.
Abstract: The Sumudu transform, whose fundamental properties are presented in this paper, is still not widely known, nor used. Having scale and unit-preserving properties, the Sumudu transform may be used to solve problems without resorting to a new frequency domain. In 2003, Belgacem et al. have shown it to be the theoretical dual to the Laplace transform, and hence ought to rival it in problem solving. Here, using the Laplace-Sumudu duality (LSD), we avail the reader with a complex formulation for the inverse Sumudu transform. Furthermore, we generalize all existing Sumudu differentiation, integration, and convolution theorems in the existing literature. We also generalize all existing Sumudu shifting theorems, and introduce new results and recurrence results, in this regard. Moreover, we use the Sumudu shift theorems to introduce a paradigm shift into the thinking of transform usage, with respect to solving differential equations, that may be unique to this transform due to its unit-preserving properties. Finally, we provide a large and more comprehensive list of Sumudu transforms of functions than is available in the literature.

299 citations

Cites methods from "Application of Sumudu transform to ..."

• ...[13] S. Weerakoon, Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology 25 (1994), no. 2, 277–283....

[...]

• ...Nevertheless, while it is well to rely on lists and tables of functions transforms, to find solutions to differential equations such as in the previous example, it is always a far superior position for practicing engineers and applied mathematicians to have available formula for the inverse transform (see Weerakoon [14])....

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• ...The Weerakoon [13] paper, showing Sumudu transform applications to partial differential equations, immediately followed Watugala’s [10] seminal work....

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• ...Once more, Watugala’s work was followed by Weerakoon [14] introducing a complex inversion formula for the Sumudu transform (see Theorem 3.1 in Section 3)....

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01 Jan 2011
TL;DR: In this article, a new integral transform namely Elzaki transform was applied to solve linear ODEs with constant coefficients, which was called ELZAKI transform and was used to solve ODE with constant ODE.
Abstract: In this paper a new integral transform namely Elzaki transform was applied to solve linear ordinary differential equations with constant coefficients.

183 citations

Journal ArticleDOI
TL;DR: The convolution theorem for the Sumudu transform of a function which can be expressed as a polynomial or a convergent infinite series is proved and its applicability demonstrated in solving convolution type integral equations as mentioned in this paper.
Abstract: The convolution theorem for the Sumudu transform of a function which can be expressed as a polynomial or a convergent infinite series is proved and its applicability demonstrated in solving convolution type integral equations.

115 citations

01 Jan 2011
TL;DR: In this article, the ELzaki transform of partial derivatives is derived, and its applicability demonstrated using four different partial differential equations, and the particular solutions of these equations are found in this paper.
Abstract: The ELzaki transform of partial derivatives is derived, and its applicability demonstrated using four different partial differential equations. In this paper we find the particular solutions of these equations.

111 citations

• ...Ezaki Example II: Consider the Laplace equation: ( ) ( ) 0 , ,0 0 , ,0 cos , , 0 xx tt t u u u x u x x x t + = = = > (7)...

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Journal ArticleDOI
TL;DR: In this paper, the Sumudu transform of a special function f (t) with a corresponding sumudu transformation F (u) has been studied and the effect of shifting the parameter t in the function f(t) by τ on the transform F(u) is analyzed.
Abstract: This note discusses the general properties of the Sumudu transform and the Sumudu transform of special functions. For any function f (t) with corresponding Sumudu transform F (u), the effect of shifting the parameter t in f (t) by τ on the Sumudu transform F (u) is found. Also obtained are the effect of the multiplication of any function f (t) by a power of t and the division of the function f (t) by t on the Sumudu transform F (u). For any periodic function f (t) with periodicity T > 0 the Sumudu transform is easily derived. Illustrations are provided with Abel's integral equation, an integro-differential equation, a dynamic system with delayed time signals and a differential dynamic system.

109 citations

References
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Book
01 Jan 1972

1,808 citations

01 Jan 1992
TL;DR: u and F (u) are no longer dummies but can be treated as replicas of t and f (t) and can be expressed using same respective units, and therefore one can check the consistency of units of a differential equation even after the Sumudu transform.
Abstract: It is possible to solve differential equations, integral equations, and control engineering problems by a transformation in which the differentiation and integration of f(t) in the t-domain is made equivalent to division and multiplication of F(u) by u in the u-domain. The new transformation which is called the Sumudu transformation possesses many interesting properties which make the visualization of the transformation process easier to a newcomer. Some of the properties of the Sumudu transformation are: (1) The unit-step function in t-domain is transformed to unity in u-domain. (2) Scaling of f (t) in t-domain is equivalent to the scaling of F (u) by the same scale factor, and this is true even for negative scale factors. (3) The limit of f (t) as t tends to zero is equal to the limit of F (u) as u tends to zero. (4) The slope of f (t) at t=0 is equal to the slope of F (u) at u = 0. Thus u and F (u) are no longer dummies but can be treated as replicas of t and f (t) and can be expressed using same respective units, and therefore one can check the consistency of units of a differential equation even after the Sumudu transform.

440 citations

Journal ArticleDOI
TL;DR: The Sumudu transform as discussed by the authors is a new integral transform that makes its visualization easier and has many interesting properties, such as: (1) the differentiation and integration in the tdomain is equivalent to division and multiplication of the transformed function F(u) by uin the udomain.
Abstract: A new integral transform called the Sumudu transform is introduced. This transform possesses many interesting properties which make its visualization easier. Some of these properties are: (1) The differentiation and integration in the t‐domain is equivalent to division and multiplication of the transformed function F(u)by uin the u‐domain. (2) The unit‐step function in the t‐domain is transformed to unity in the u‐domain. (3) Scaling of the function f(t)in the t‐domain is equivalent to scaling of F(u) in the u‐domain by the same scale factor. (4) The limit of f(t) as ttends to zero is equal to the limit of F(u)as utends to zero. (5) For several cases, the limit of F(t)as ttends to infinity is the same as the limit of F(u)as u tends to infinity. (6) The slope of the function f(t) at t =0is the same as the slope of F(u) at u = 0. Hence uand F(u)are no longer dummies but can be treated as replicas of tand f(t).It is even possible to express uand F(u)using the units of tand f(t) respectively.

400 citations

Book
30 Nov 1971
TL;DR: In this paper, an intermediate-level text on the use of integral transforms in applied mathematics and engineering is presented, which is divided into five parts covering integral transform pairs, the Laplace transform, Fourier transforms, Hankel transform, and finite Fourier transform.
Abstract: An intermediate-level text on the use of integral transforms in applied mathematics and engineering. Existing works either cover the subject in more elementary form or are advanced treatises. In a very lucid style the author deals with the use of this important mathematical tool to solve ordinary and partial differential equations in problems in electrical circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. The book is divided into five parts covering integral transform pairs, the Laplace transform, Fourier transforms, Hankel transforms, and finite Fourier transforms. A basic knowledge of complex variables and elementary differential equations is assumed. There are many exercises and examples drawn from the above fields, tables of the transform pairs needed in the text, and a glossary of terms with which the student may be unfamiliar. For the student who seeks further background on the subject, an annotated bibliography is provided.

44 citations

MonographDOI
01 Jan 1971

24 citations